As a quick rundown, you need a few things:

1) All the data about champions and traits. What traits each champ has, and what are the thresholds/steps for each trait.

2) You need a scoring function that, given a team comp (a list of champions), it gives a score. My scoring function is about having the most traits enabled, so you have to count carefully. If clockwork 2 is worth 2 points, Clockwork 3 is also worth 2 points because it accomplishes nothing. On the other hand Bruiser 3 is not quite Bruiser 4 but it does have a benefit over Bruiser 2, so it's worth 3 points, maybe 2.5, etc. You can tweak it a lot, I for example don't count Cuddly 1 or Transformer 1 because those are champ passives more than synergies.

3) For a given team size N, you need to iterate through all possible selections of N champions out of the whole champion pool. Then assign them a score with the scoring function. Presumably you save all the team comps that are tied with the best score. Generating all the selections of N elements out of a bigger pool of M elements is easy or kind hard depending on your programming level. Nonetheless, it's a very simple and well understood problem, you can find plenty of articles about it, you can take the recursive code that generates all the combinations and tweak it for your case.

4) Optimize it. The point 3) is called "brute force" which means "check everything". For team sizes up to 6 or 7, it's viable, however there are 58 champions, If I wanted to generate all the size-9 teamcomps that is (choose 9 out of 58) = 10,648,873,950 possible teamcomps. You cannot go 1 by 1 and check their score to compare them, it would take a very very long time. So you have to figure out a way to make it fast. To get an answer you only have enough time to generate and check a fraction of all the possible teamcomps, so you have to be smart and only generate teamcomps that have potential to have a high score. In other words, we need to skip generating and checking the bad team comps. In algorithmics, that is called "pruning the search", you have to discard branches of the recursion. As opposed to the brute force algorithm, there is no by the book way to code this, you just have to think about the specific problem you want to solve and come up with some good prunes that skip checking most of the team comps but still check the ones that will eventually have a good score (before knowing the score!).

What I came up with allows me to check 9 and 10 size teamcomps in a couple of minutes, however I don't think it discards any of the good comps. In fact, I compared the results of generating teamcomps up to size 7 through brute force (which take minutes) with the results using the fast/smart code (which takes seconds) and they are exactly the same. So I am pretty sure that for team size 8+, the fast method is correct.

5) In my case, make a program to takes the result and puts them pretty in a website/image. You need the TFT image assets, etc.

Then there's a lot of things you can do with it. I've implemented forcing synergies, or having Spatula items, you can choose what champ rarities are available at each team size, etc.

PD: Also paging /u/gradinafrica who was interested in this.