OK, I have several suggestions, but they are based on limited data. I am not quite sure what you are describing.
First, this is an I told you so moment. Always back up the .enc file nightly, and ALWAYS automatically back up the entire compute nightly. OK, done with that.
If I understand correctly, you had 6.0.9 on this computer and it was working fine. You then tried to download 6.0.10, and the computer froze over and over. My first question: So, it was just the download. It could have been downloading Registry Cleaner. Had nothing to do with AC?
So, now you can't use that computer due to this freezing no matter what you do?
So, this brings up several options or ideas:
Does the computer just freeze even if you don't use it? If so can you get to the information from another computer? Can you take a computer and use a crossover cable and connect directly to the bad computer's database?
Can you do a system restore on the computer? It sounds like the download corrupted a driver.
Boot into Safe Mode with networking. With only the basic drivers, it will be less likely to crash, and unneeded services such as SQL Server\AmazingCharts would most likely be turned off.
Can you do a system restore from the Safe Mode entry prompt? When you boot into Safe Mode, you get a chance to retore (No) or continue (Yes).
If you can get to the data via Safe Mode, you can make a backup of the data (it's still possible). Back it up locally and to another external site. Then you would be golden.
If not, use the Amazing Utilities to Detach and Move the databases to another location on the computer and then copy and paste them to a flash drive or another computer. If you feel more comfortable, you can just detach them and copy them so that a copy stays with the program. If you aren't atached to the SQL Server engine due to the service being off, you can just move them anyway.
There is no way to know if it is the hard drive causing the crashes, but we do know that data is on the hard drive, so if all else fails, remove the drive and put it into another computer.
If it is only that the computer is constantly freezing/crashing, a local computer shop should be able to bench it and see what s casing the freezes. It could be as simple as a video card or as bad as a processor or motherboard. But, they have more tools to tell.
If you are really desperate, then acquring the informtion directly from SQL using SQL Server Management Studio may be an option. The hard part would be the remoting into the computer as it freezes. If you go this router, I would once again, use my SQL Expert who is the only one I know that could do it. Obviously, there are thousands, but this guy is good. But, I know you won't, because nobody does.

If you have System Restore, I would start there. And, if you can't even get to it, you can access it via Safe Mode.