Buy all weapon upgrades in the upgrade store. You'll need 44,000 total style points in order to buy every one.
If you want to get both this and the ability upgrades in one playthrough (in order to get "Hey Big Spender!"), you'll need to collect as many points as you can, because points are scarce. Some tips for maximizing your score (stolen from my guide to "Keep Your Balance"):
1. Keep your multiplier up! This is especially important during spawn door sections. Just keep shooting.
2. Don't destroy spawn doors until they stop spawning enemies.
3. Killing bad guys while sliding after a dive will get you twice as many style points as when you kill them normally.
4. Destroy all crates to make sure you get any bonus style points.
The achievement will unlock as soon as you commit to buying the upgrades.