The game starts skipping as soon as the gameplay begins (after the intro). The framerate stays smooth, it's something that happens to the character. When I try to, for example, look around, the whole screen goes nuts, it starts acting as if mouse was borked. The same thing happens with the input from the keyboard, the character moves too far away or too close, his animation skips, he flickers and other strange things happen.
I'm on Win 7 64, Athlon 64 X2, 4 gigs of ram and Radeon 4870.
Any ideas?

Have you tried the general trouble shooting steps for old games?
Have you tried compatibility settings? If not to do this just right click the games shortcut and go to properties, then the compatibility tab. Try running it under win xp sp3 and setting it to run as administrator.
Also make sure that your ati control panel all the gobal settings are set to application controlled so you are not getting any conflict with that. Unsure of how to do this on with an ati card but if you find the control panel the options should be in there.
Also you can try running the game on a single core. To do this just load the game up and then ctrl-alt-del and bring up the task manager. Find the game in the processes tab and right click it. Then select affinity settings and set the game to only run on core0.

Yes, I did.
I tried updating Catalysts, changing Affinity and running it in Compatibility Mode. Nothing helps. Actually setting it to run under Win XP SP3 makes things much worse.

Might try the AMD Dual Core Optimizer as it helps with some graphic problems. It installs and runs in the background, there is no user interface. It can be uninstalled in the control panel.
I have been running it for several months with no problem. As always, make a system restore point prior to installing. . . . =)

I have it installed. Doesn't help.

Only other suggestion I can think of is to install outside of the Program Files directory, make sure you install by right clicking and clicking "Run as an administrator". After install, right click the game shortcut, click Properties, click Compatibility tab, uncheck "Run this program in compatibility mode for" to begin with and check "Run this program as an administrator" Make sure to uninstall and delete any leftover directories under Program Files\GOG.com\[game name] before reinstalling
Sorry I can't be of more help, haven't installed it yet so . . . =)
Edit: I would turn UAC off until the install finishes up.

I guess as its an old game maybe rolling back your graphics drivers might work. Its a bit of a pain in the ass to do for a single game but if it works at least you have narrowed down whats causing the problem.

I too have some skipping, turning of V sync helped a little, going to try to turn off hardware sound as well. Setting affinity didn't really seem to do anything. Game is running TOO fast it seems, and sometimes like, 10 frames all buffer and release at once. That is what it seems like at least.

I have Vista 32 bit on a laptop and the game installs/plays flawlessly. Smooth framerates and the music is excellent.

Bummer . . .
Did you look at this page?
I would open a help ticket with GOG support if you have not already done so.

Reading the Enclave review by dnoelsmith, I found the following comment -
Note that if you think your graphics card can handle OpenGL, you can modify the environments.cfg file and change the following line to use OpenGL instead of Direct3d:
Change this line VID_RENDER=Direct3D to VID_RENDER=OpenGL
A strange place to find this but it may be worth a try.

The game seems to use code which is affected by the speed of modern computers. On my PC it sometimes speeds up noticeably.

Could try a cpu limiter then, not sure which ones are good or safe though.

Unfortunately it doesn't help. OpenGL just makes the gamma less brighter :(.