If possible, you should overclock the CPU to give you better performance for CPU dependent games. Overall, it is not a bad CPU given it's age. However, it can bottleneck you in some games.
The Radeon HD 6670 is pretty weak. When it 1st came out it was considered a mainstream graphics card in terms of performance. It now falls under "budget" graphics card performance since many more powerful graphics card have been released since then. As an example, the Radeon HD 7750 is currently considered a mainstream graphics card; it's performance is equal to the Radeon HD 6770 which is around 40% more powerful than the Radeon HD 6670; that's a "guess-timate" I am pulling from memory.
To give you an idea of the Q6600's performance level in current games with a powerful GPU below are several CPU performance charts. The Q6600 would be a marginally slower than the Athlon II X4 640 which means it will be at the bottom of the charts. In Call of Duty: Black Ops II, if paired with a Radeon HD 7970 you would get in the high 70's like 78 FPS or 79 FPS. Pretty good since most people only have a 60Hz LCD monitor which means the monitor can only display at most 60 frames per second. But the benchmark shows that more current CPUs will allows for higher FPS, even though a 60Hz monitor will not be able to display them. You would need a 120Hz monitor if you want to be able to see more than 60 FPS; up to a max of 120 FPS; assuming the graphics card is powerful enough.
BioShock Infinite represents a more demanding game, but based on the performance chart with GTX 680 a Q6600 should be able to get low 50's FPS. A more powerful CPU will give you better frame rates, but from the looks of the charts the bottleneck of the Q6600 isn't too bad.
Moving on to more demanding like Far Cry 3, you begin to see the performance gap between the Q6600 (Athon II X4 640) and more powerful CPUs with a Radeon HD 7970. The bottleneck is considered significant enough to be noticeable if the Athlon II X4 640 was played side-by-side along with i5-3470.
Hitman: Absolution further widens the performance, however, most games are not as demanding as Hitman, but this is something you should consider nevertheless. The Athlon II X4 640 get 29 FPS with the Radeon HD 7970, so the Q6600 may get 27 or 28 FPS. Notice how even the dual core i3-3220 give significantly better performance; but in general you will want more than 2 cores in the CPU for a gaming rig. In this game Intel CPUs more or less dominates.
Lastly, Crysis 3 is brutal on any system. Note that in the benchmark below with the GTX 680 the graphics quality is only set to medium, not high like all the other games. You can see how poorly your Q6600 will perform in this game. However, Crysis 3 is an exception, not the norm. The vast majority of games are not as demanding as this games. Most games will be somewhere in between BioShock Infinite and Far Cry 3; at least in my opinion.
http://www.techspot.com/review/601-black-ops-2-performance/page5.html
http://www.techspot.com/review/655-bioshock-infinite-performance/page5.html
http://www.techspot.com/review/615-far-cry-3-performance/page6.html
http://www.techspot.com/review/608-hitman-absolution-performance-benchmarks/page6.html
http://www.techspot.com/review/642-crysis-3-performance/page6.html