I'm not sure exactly which kind of proof you're looking for, but a proof by induction using Gröbner bases is presented in Chapter 15 of Eisenbud's *Commutative Algebra*, see Corollary 15.11 specifically. 

There's also a more abstract homological proof in Chapter 19, see this related question: http://mathoverflow.net/questions/26783/best-exposition-of-the-proof-of-the-hilbert-syzygy-theorem-by-eilenberg-cartan

The first proof is closer in spirit with Hilbert's original proof.