The joint distribution of X and p is...

where 0<p<1 and x=1,2,3,...

In order to obtain f(p|X) you need the marginal of X.

NOW use the beta density http://en.wikipedia.org/wiki/Beta_distribution

where x=1,2,3,...

This is a telescoping series that does sum to one.

Now divide and you should have that conditional distribution.