How to derive association from a regression model?
I think NN is a little problematic here, because you run into the bias-variance-tradeoff. NN are really good in producing estimates with low variance. However, if you look for a causal influence of $x$ on some $y$, you would choose an unbiased estimator (with higher variance). Options are OLS or Lasso/Ridge estimators. Especially with DNA information, Lasso is a good choice since you may face a problem with (relatively) high dimensionality.

If you really want to explore feature importance in a NN setting, I guess you need to resort to stepwise forward/backward techniques similar to the method as used for feature selection. See here and here for some background. To my best knowledge Keras does not come with a predefined function for the problem.