How to draw a hyperplane using the weights calculated
I think a better to way to think about a neural net as you have it written is that it is a function.

$$f_{nn} : \mathbb{R}^n \rightarrow \{0, 1\}$$

which is generally a non-linear decision boundary. You did not indicate your network structure and activation function. If your data is linearly separable then a single layer perceptron will define weights for you in the input layer, plus a bias input node for the intercept. 

If you specifically want a separating hyperplane you could consider support vector machines. See here.

https://stackoverflow.com/a/8021219/431895