I have no experience with the 770 (I have Rem 700's, and Winchester 70's), but what eagl wrote about matters a WHOLE LOT. If you don't/can't use and experiment with reloads, it could easily pay big dividends to go out and buy several different loads/brands to test. I've seen that take a horrendously ugly group and fix it immediately.
Another point worth mentioning (because I see it almost every year with people sighting rifles before deer season) is your rest set-up. You can take a perfect rifle, perfect scope, and perfectly matched rounds and make them shoot bad by having an ineffective rest, or in particular a poor rest material. In my case, I took a beautifully-grouping rifle and caused myself all sorts of headaches by padding the front and rear rests with a bit of 1" foam (it seemed like a good idea at the time). My groups went bad immediately, but i didn't realize the rest was the cause until around 50 rounds later (and a lot of time spent investigating my scope mounts, scope, cleaning (I thought maybe I had fouling of some sort causing problems). I even made up several new reload combinations... No luck... Then I took the foam off, and bingo, good groups again.
I've seen the same effect when people yank the foam out of a case to use as a rest at the range, or rolled up towels, or a soft case folded in half, etc...
I brace both the fore stock and toe of my rifles when sighting them in, because I don't want any movement at all.
You may have a good rest already (I find good ol' sand bags to be best/easiest), but if not, I'd look into that first, since it's such an easy and overlooked component. It may not be a problem for you though, because you mention using a vice to hold the rifle. Depending on the vise, using it may have already eliminated the scope and mounts from the equation. If' that's so, I'd go back to what eagl mentions.