If your hated then it's probably too bad to reverse.
No way to. The only way to improve your reputation is to equip faction armor (so that you don't get shot on sight) and do some quests for the faction, to get reputation to Unpredictable or better.
There's still no way to do quests, because guards and rangers see past faction armor. Approaching rangers or legion praetorian guards will get you caught.
If you haven't gone to the Strip yet your faction status for the NCR and legion will reset to neutral when you get there (Crocker/Vulpes Inculta will contact you and then you'll know). Otherwise you've pretty well decided through your actions. You can set yourself a new status through the console, but that may prove buggy.