How I know which gene is a good predictor in this neural network or not?
Yes, there is a formal solution. You want a heat map of the layers weighting and this is the information sought. It is more difficult to interpret than other supervised learning algorithms, but that is the nature of neural networks. 

I know Sci-test Learn (Python), but I don't know R for neural networks, so I can't advise you on the code, only that it will be a few lines of code and a graphical (relative) output. I imagine Tensorflow will have a really brilliant heat map layer weighting.