The PHB states that a wizard add 1d6 + constitution mod per level after first level. Would I subtract hit points when I level up if I have a negative constitution modifier and roll badly?
Your hit points will never go down from leveling up. You add 1d6 (or 4), and apply your Con modifier, but the minimum of that is 1, so even if your Con modifier is -3, and you roll a 2, you add 1 to your hit point maximum for leveling up. I don't know if that has been clarified in the official errata, but at a minimum there's a Mearls tweet to that effect: https://twitter.com/mikemearls/status/582601826031132673