So I got this computer from a friend and right off the bat ran CoreTemp. During a game the temp was listed at critical as it crested over 86 degrees celcius. When the computer is in sleep mode and just turned on it sits around 30 Celcius. Then just operating brings it up to between 45-60. I am afraid to run a game anymore since it gave me a critical heat warning.
There are 3 fans in there running on medium as well as a sizeable heat sink.
It could be the thermal paste has dried up and is now more or less powder. So my 1st recommendation is to replace the thermal paste. I recommend using Arctic Silver 5, it's been around for awhile, but it's still very effective.
Turn on the PC for awhile, then shut it down before removing the heatsink. Heating up the thermal paste should make it easier to remove the heatsink.
Download a program called CPU-Z. Then run an intensive CPU program. Check the Core Speed. The E6600 at stock speed runs at 2.4GHz, anything higher means it is overclocked. You can start a game in window mode or download a program called Super Pi to stress your CPU.
As for a heatsink recommendation, it would be nice to know which one you currently have. You can simply ask your friend. If it is a good heatsink, then all you really need to do is replace the thermal paste.
I don't particularly like either one a lot since when I look for heatskink, I want something that is both very effective and also nearly dead silent. The only that fit's that those criterias is the Noctua NH-D14 120mm & 140mm SSO CPU Cooler ($90), but it is huge (may not fit in your case) and it is quite expensive for an old PC.
The good news is that it is compatible with Intel's current socket 1155 CPUs (Sandy Bridge and the upcoming Ivy Bridge). However, if you wait too long to upgrade, then it will be difficult to find new parts to build around a socket 1155 CPU because when Intel releases their next CPU; Haswell (socket 1150); they will basically stop producing socket 1155 CPUs.
Actually Intel in all their naming scheme wisdom released 2 different processors called E6600. The first is a 65nm Core2Duo Conroe core and is clocked at 2.4Ghz. The second is a 45nm Wolfdale core called the Pentium E6600 and is clocked at 3.06Ghz. Might be confusing if you are trying to tell if you are overclocked.