Nullifying Nightmare, Blaze Boy, Greenery Giant, Tidehollow Tyrant, and . . . yeah, okay, Phoenix Paladin.

(I was on the fence about whether the last spot should go to Paladin or Ranger, but when I saw Measure's answer I decided to let hipsterism be the deciding factor.)

There seems to be a rock-paper-scissors thing going on here: Earthy fighters have an advantage over Watery fighters, Watery fighters have an advantage over Flamey fighters, and Flamey fighters - kinda, sorta, unreliably - have an advantage over Earthy fighters. (And the Nightmare has an advantage over everyone.)

This is relevant because 3/5 of the opposing team is Earthy fighters, including Greenery Giant, who has strength that rivals the Nightmare, and whose presence on a team predicts a ~60% chance of victory.

Teams which are slanted too heavily towards a given element have an extremely low win rate. I can't tell to what extent this is because losing the rock-paper-scissors game hurts you more than winning it helps, and to what extent balance is inherently valuable, so I'm playing it safe and not building an entire team of firestarters (also, there are only two Flamey fighters with non-terrible win/loss ratios).

I infer from the format of the alternative list that - absent an extremely tricky fakeout - position doesn't matter: A+B+C+D+E is equivalent to E+D+C+B+A.

Different fighters are used with very different frequencies, but this sampling bias doesn't seem to affect my analysis much.

Eyeballing the correlation matrix, it looks like teams are thrown together randomly; no pairs that always show up together, etc. This makes things much simpler, since I can be confident that (for example) GG's apparent power isn't just because people keep using him alongside NN (or vice versa).

There's a random element here. Existence proof: A+B+C+S+V vs A+E+I+T+V happened twice with different outcomes. Given this, I'd want to push Cloud Lightning Gaming to have the match be best-of-five, to decrease randomness' relevance to the outcome.

I appreciate the omission of letters that would let us (accidentally or otherwise) spell out common swearwords.

DM'd

Spoiler protection.

Team colour doesn't appear to have any meaningful imapct on the chances of winning.

There are onl2 games where the opponents combination was played which is too small to draw any conclusions from, however there are 352 games where a team has had 4/5 of these characters. 

Teams with Greenery Giant or Nullifying Nightmare in them seem to do unusually well against Dire Druid (Nullifying nightmare does slightly better)

Greenery Giant looks very strong there is no other character that has been part of a winning team > 50% of the time against teams which she has been in, though Lanslide Lord comes close

Nullifying Nightmare and Tidehollow Tyrant seem to do best against Phoenix Paladin

Quartz Questant is also fairly strong, but teams with Blaze Boy in tend to win slightly more often than teams with Quartz Questant in. Greenery Giant and Nullyfing Nightmare also do very well. Rock-n-Roll Ranger however does better than all except Greenery Giant

Against teams with Tidehollow Tyrant in them teams with Greenery Giant, and Rock-n-Roll Ranger do best.

 Which suggests the following line up:
 - Greenery Giant
- Landslide Lord
- Nullifying Nightmare
- Rock-n-Roll Ranger
- Tidehollow Tyrant

 Both times this combination was used it won.
Teams with at least 4 of those characters won 77 percent of times, which suggests this is a strong combination.

I will therefore use the following for both the regular and PvP teams:

 - Greenery Giant
- Landslide Lord
- Nullifying Nightmare
- Rock-n-Roll Ranger
- Tidehollow Tyrant

ABGLP

BGNPT

made a random-forest model, tried all possible hands against DGPQT for the regular team, and ran some tournaments using that model to adjudicate for the PvP team. I haven't made any attempt to use my brain to find any patterns, and it's entirely possible that I've missed important things.

The random forest model was built using the RandomForestClassifier class in the scikit-learn library for Python. It uses an ensemble of size 1000 and other parameters had their default values. It's rather slow so for the tournament I used a cheaper random forest model. For PvP, what I did at first was to run 10 sub-tournaments with 30 players in each, initially selected at random. Each tournament is played round-robin and then three badly-performing hands are replaced by mutated versions of three well-performing hands. Scores in the tournament are not pure win/loss but are the total of the win probabilities according to the random forest model. After every 10 rounds in each tournament, I play another tournament using 3 randomly chosen players from each sub-tournament, again doing the replacing thing (so we get a little bit of mixing between sub-tournaments, but hopefully not so much that all diversity is lost too quickly). I ran a couple of hundred iterations of this and it seemed to have more or less converged. To choose my final PvP hand, I ran a single tournament on the best three hands from each subtournament using the more expensive and hopefully better random-forest model. -- But afterwards I thought better of this and decided instead to use the "multiplicative weights update" approach: just take all the possible teams and give each an initial "weight" of 1, then repeatedly play each against a random opponent and nudge its weight up or down according to the result, and choose the random opponents proportionally to the weights. Exactly what team comes out on top depends on exactly how you do the weight-nudging.

Blaze Boy
Greenery Giant
Maelstrom Mage
Phoenix Paladin
Rock-n-Roll Ranger

I'm not super happy about it, because I know nothing of how the matchups actually work. This is just a product of a kinda ad-hoc value function. How well do individuals do? Multiply. How well do teams do, relative to the product of their members? How well do individuals on a team do against the known opponents, pairwise? Okay try all teams, combine their "vs opponents" scores with "team coherence" scores, pick the max.

Blaze Boy, Greenery Giant, Nullifying Nightmare, Rock-n-Roll Ranger, Tidehollow Tyrant

Blaze Boy, Greenery Giant, Landslide Lord, Rock-n-Roll Ranger, Phoenix Paladin

I looked at each champion's win-rate against every other champion. Then I took the 5 champions on their team and looked which champions had the highest mean winrate against all champions on the opponent's team. 

Coincidentally, there are exactly 5 champions that have a >50% winrate by this criterion. So I'm bringing all 5 of them as my team.

The same as my regular team, embarrassingly enough.

 Greenery Giant, Rock-n-Roll Ranger, Landslide Lord, Blaze Boy, Tidehollow Tyrant

I decided to iterate the process I used to beat the opponent's team. And discovered that the 5 best champions against that team were, in fact, that team.

I feel like this shouldn't happen. Maybe I'll look more at this tomorrow and change my mind.

Using python I conducted a few different analyses:

Proportion of character wins vs other characters:
Proportion of character wins when paired with other characters:

With these I gave each possible team a score, equal to the sum over characters (sum over enemy characters(proportion of wins for said character) + sum over other teammates(proportion of wins when paired with said character)), and the highest scoring team was:

Rock-n-Roll Ranger, Blaze Boy, Nullifying Nightmare, Greenery Giant, Tidehollow Tyrant

This was much more pleasant than using Excel! I think I might try and learn R or some other dedicated statistical language for the next one.

PvP team (without having the time to estimate anything about my enemies' teams so highly likely to get countered) has actually come out the same. There's a good chance something is up with my analysis or my method is too biased towards synergistic teams.

Tidehollow Tyrant, Rock-n-Roll Ranger, Nullifying Nightmare, Greenery Giant, Blaze Boy

the water, fire and earth groups remarked on by abstractapplic are, it seems to me, more easily noticeable via their (anti-)synergies than via their counters, which seem somewhat hit and miss, presumably due to more individual countering effects. The groups are:

Fire: BFIOPV

Water: ACMSTW

Earth: DEGLQR

and Nullifying Nightmare is separate from each.

The enemy team is, recall, DGPQT

My initial pick was BGNRT was based on green side counters (imagining the enemy on the blue team). GNT are generically strong, R counters DQT on the enemy team, and B counters D and P and (weakly) Q, is not as bad against G as most heroes, and is not as badly stomped by T than other fire characters.

The synergies did not seem important enough to affect the team selection particularly much.

Maxwell used 350 training steps, based on green and assuming the enemy to be blue. His code outputted AGLNP as the best counter to the target enemy team DGPQT.

My remarks:

The validation is still improving when Maxwell stops training in the code as supplied in that post. So I increased the number of training steps. The risk is that it will just overfit more, but I assume that if that was a big problem it would make the validation worse? 

Changing to 1000 training steps results in the code outputting AGLNO as the best counter to the enemy team, with AGLNP as the fourth pick.

The problem as supplied by aphyer does not seem to specify as to whether we are Green or Blue. So we should probably be prepared for both (and this especially applies for PVP). I also wondered if the apparent differences were due to random chance. So, I created a flipped version of the initial csv file (changing all green characters to blue and vice versa, and also the wins). Maxwell's code run on this flipped version outputted LOPRT as the best counter to DGPQT  - quite the change! 

I also created a merged csv file from the original and flipped files and 2 cut csv files from the merged file which each contain some entries from the original and some from the flipped file (but no overlap between each other). Validation scores on the cut csv files were worse than on the flipped and original csv files, confirming that side does matter (unless I misimplemented the flip, which would embarrassingly invalidate all my subsequent analysis...). However, even though side does matter, since we don't know what side we are on, I figure we are mainly interested in averaged data so used the merged file anyway. I also increased the number of steps to 1500 for the merged model and doubled the cutoff indices for the validation (because 2x the data).

The code now outputs BGLPR as the best counter to DGPQT. So, that's my choice (unless I change it later). It also happens to be one off from GuySrinivasan's choice (who picked M instead of L).

For PVP, against the average team the code now recommends BGNOT. However, I'd like to modify it to be more specialized against likely opponent teams. This will take some time, possibly more than I can spare, since I have not used R before. If I don't have time to adjust it, I will go with BGNOT.

edit: now switching to ABGOT as the PVP team. According to  the code this counters my above choice, as well as gjm's pick of BGNPT (sorry gjm). Also seems to do OK against others who have published PVP teams (a good reason not to publish them). Attempts to get a better value estimator for PVP teams and apply it to all possible choices have thus far been thwarted by my unfamiliarity with R.

Current choices:

 BGLPR for main answer

ABGOT as PVP team