The 5770 is 4 years old. If money isn't an issue, upgrading to a new videocard could be a good investment. As I said before, if you have limited budget, it's better to invest your money in a videocard. You can get a decent videocard for $200 (e.g. an nVidia gtx560ti). Or a great one for a lot more (e.g. the new gtx670 for ~$400 will be a very good choice). Those cards will give you higher framerates, and will allow you to use more eyecandy. I think the gtx560ti is a decent match for your system. The gtx670 might be held back a bit too much by your system. You can always buy a new CPU (+mobo+ram) next year or whenever you feel like it.
You could buy an ATI card in stead of an nVidia card. But from what I've read, they seem to have a bit more issues with certain games than nVidia card. For pure price/performance, I don't think there's much difference between the 2 brands. Except that in some games, one brand runs better than the other. E.g. I think nVidia cards have slightly better framerates in Skyrim that AMD card. On the other hand, AMD card tend to have a bit more vram. And thus if you run Skyrim with lots of addons, texture packs, high resolution, etc, then an AMD card (with more vram) might be better. It depends ....
Of course, if money isn't an issue, then buying a new CPU, motherboard, ram and videocard would give the best results.

Easy.

But it would set you back $220 for an i5-3570K, $100 for a z77 motherboard, $50 for 8GB of ram, and either $200 for a gtx560ti, or $400 for a gtx670. That's $600-$800, but you would have an awesome gaming system that would again last a couple of years.
But if I were you, I'd try to see if I could fix that problem first. That would save you some money.
Get GPU-Z.
http://www.techpowerup.com/downloads/SysInfo/GPU-Z/
Start GPU-Z. Start the task manager (performance tab to watch cpu usage). Play Skyrim. Alt-tab. What is your CPU usage ? GPU usage ? Ram and vram ? (Check 2nd tab (sensors) in GPU-Z for vram and gpu usage).
I am sure there are better tools (I used nvidia inspector before, but I bet that doesn't work with AMD cards).