I'm not sure where you are starting in terms of background and references, but the standard short book for such questions is Serre's *Linear Representations of Finite Groups* (Springer GTM 42, a good English translation by Len Scott of older lecture notes dating back in their first version to the 1960s). Part III mixes the ordinary and modular theories (where the characteristic is 0 or prime).

For the "easy" case where the prime doesn't divide the group order, see Serre's 15.5. Of course, this is based on a lot of previous general theory, so you may prefer a more direct argument along the lines suggested in Ben's comment. Either way, you have to start in characteristic 0 with a matrix representation, then find a suitable basis for reduction modulo $p$. Starting with an irreducible representation (over a big enough field such as $\mathbb{C}$) then yields an irreducible representation in characteristic $p$. (But there is some theory required for this process.)

The case when $p$ does divide the group order is fairly subtle, which shows up in the more sophisticated tone of Serre's third part. (The old 1962 book by Curtis and Reiner is more down-to-earth, but doesn't get to this topic until the very end.) The short answer to your question here is that lots of things can happen, as Richard Brauer and his followers have shown in detail. The dimensions of irreducible representations in characteristic $p$ usualy won't divide the group order and have nothing obvious to do with dimensions in characteristic 0. But there are subtle connections, still being worked out for simple groups in particular. Only narrowly focused questions are likely to have clear answers.