The Pats cut vets when either three things happen: the player is no longer playing at a top level (Logan Mankins was horrible last season), the player puts himself ahead of the team (Asante Samuel) or they can't afford to give him a contract extension (Richard Seymour) cause they have other players to resign. If you really want to build and draft just like Belichick does read the books written about him; Patriot Reign, Education of a Coach and War Room. These will give you a small insight into the mind of a football genius.
When it comes to the draft specifically you need to throw normal convention out the window. Belichick doesn't draft like any other coach/GM in the league. If he likes a player he will draft him as soon as he can even if he projects the guy to be nothing more than a special teamer he will spend a 2nd or 3rd round pick on him. To him as long as a player can contribute to the team in some way he is a valuable draft pick. Also resist the urge to sign the big name free agents and instead find a cheaper alternative that is close to the same skill level. For instance instead of the 1000 yd WR who wants $10-12 million, sign the 700 yd WR who wants $5-6 million. The difference can be easily made up especially when you have an elite QB.