I have the HD 5670 too. Skyrim has been playing fine for me at 1920x1080 at mostly high settings, some medium, and everything else set to none (ex. AA). It runs pretty well. The optimized graphics settings that Skyrim automatically setects for my system makes the game play like a vey fast slide show. So those are the graphics settings (mine) that you should aim for. Now onto the question...
Since the 1.5 update (few weeks ago[1-2 months]), I have been having the same problem. It loads, I play for a few seconds, and then it crashes. Never used to happen before then. But then again AMD updates Catalyst Control Center every month, so that could of been it. I actually just installed the brand new update for CCC, so Im about to try it. Hopefully for the both of us, good luck to me!