This is a very interesting problem that can be approached in a lot of different ways!
Even though you are asking for a neural network, I will apply my own experience with Bayesian networks as I think they are more than suitable for this task (they might be even the best choice).
You could start of with a small seed of highly popular users which you can label as gamers/not gamers. Afterwards, using some threshold (which should vary in proportion to the number of users which have been labelled P(θ = X | labelledUsers = 100)), you should decide whether a particular user is a gamer or not (I would start off with people that follow a lot of "twitterers" to maximize the chances of following one of your "seed".
To help you find the previously mentioned probability, you can also help yourself with a test set to validate the probability. I hope this was helpful even though it doesn't use a neural network. BTW, you could also model P(θ = X | labelledUsers = 100) with a beta distribution. Using this technique, the algorithm would choose how much to weigh each feature using a PDF and you could just feed it as many features as you like without bothering about how relevant they are. As an extra, I wholeheartedly encourage you to buy/download John Kruschke's book on
Doing Bayesian Data Analysis (A tutorial with R and BUGS), it will give you an excellent insight into data analysis for solving problems like this one.
In case you still want to use ANNs, I would recommend learning the parameters (the weights) for the neural network in a special environment (which you have to manually label) and using features such as the number of people following, ratio: gamers/all, you can try to create a computer vision algorithm for the images he posts (gaming vs non gaming). The list is infinite! Still, you should be careful not to include too many random (meaningless) features otherwise it might cause underfitting!
If you would also like details on which programming language would be best for this, I would recommend R or Matlab (my personal choice even though R comes with more premade functions) for developing the algorithm and Fortran or C for a final exportable version (if you wished to do so).