"Neural network" is largely a buzzword, real game AI is a lot of different things, but rarely something that fits the term thoroughly.
The closest thing that can broadly be applied to games is automatic optimization of magic constants, that is when you have written a reasonable AI for your game, you can take all the potentially different constants of that AI, put them on a list, and have the AI play against itself with different values of those constants to determine which set is better. You can then take the winning set, apply some further changes, let the changed set play against the original etc. Depending on how fast you can play a game you might end up running through millions or billions of games in the process.
The process itself has to be crafted carefully, if you adjust the numbers too little it will be too slow, if you adjust them too much you'll almost never get anything better. If you don't play enough games with a given set of AIs the result might be too random. But do in general make sure that there is some random behaviour or you risk making an AI that only plays well against the previous iteration. Having a broad set of different enemies to test against may also be a good idea, even if they are objectively not very good.
And of course this whole process won't save you if you haven't made an AI that is suitable for it in the first place.
If you want to test it out on Tic Tac Toe, write an algorithm that for each possible move generate a number based on some properties of that spot. Then choose a move randomly with greater chance for choosing a move with a high number. If don't include the chance you won't train for countering a lot of the possible moves.