Prove that my NP-complete problem is strong
You've asked two questions.  To answer the first question:

Not quite right.  It depends on the running time of your dynamic programming algorithm.  To show that the problem is weakly NP-complete, you should show that there is an algorithm whose running time is polynomial if the input is represented in unary.  So, you'll first need to prove that your dynamic programming algorithm has that property.

To answer the second question:

If in doubt, go back to the definition.  Wikipedia has a page with a definition of strongly NP-complete; it says


  A problem is said to be strongly NP-complete (NP-complete in the strong sense), if it remains so even when all of its numerical parameters are bounded by a polynomial in the length of the input.


So, that's what you need to prove.  It's hard to say much more without knowing what the specific problem is.