As it is for ANYONE, the bottleneck usually exists in the CPU or GPU but it varies by the GAME quite a bit.
With a weaker CPU the bottleneck is more likely to be the CPU, but there will be times when it is not. To get a rough idea, let's look at the i7-920 approx processing power (adjust results if you overclock... which you should likely):
https://cpubenchmark.net/cpu.php?cpu=Intel+Core+i7-920+%40+2.67GHz&id=834
That's best compared to the FX-4300:
https://www.cpubenchmark.net/cpu.php?cpu=AMD+FX-4300+Quad-Core
If we look at the SINGLE core it's actually quite a bit weaker. Just over 80%. (Note the TOTAL is similar but that's because the i7-920 has hyperthreading. Currently very few games benefit from hyperthreading on a 4-core CPU). The FX-4300 is already a bottleneck by over 25% in some games so yes, it will be a very SIGNIFICANT bottleneneck at times.
TOMB RAIDER:
http://www.techspot.com/review/645-tomb-raider-performance/page5.html
It would appear that CPU performance here doesn't have much impact. In reality it will matter more than this graph suggests as we'll get more stutter and low FPS dips but this at least shows the CPU doesn't matter nearly as much in some games.
FALLOUT 4:
It's mostly an outdated engine, but possibly a game you'll play. It's very CPU intensive:
http://www.gamersnexus.net/game-bench/2182-fallout-4-cpu-benchmark-huge-performance-difference
*We established that you're about 83% the perf of an FX-4300 so that's about 40FPS in the 1080p results if everything was setup the same (same GPU, settings etc) whereas the best CPU got over 2x the performance there.
Starcraft 2?
If you play that game it's slightly unique as it uses two cores of the CPU so especially during heavy battles the CPU single thread performance dictates the benchmark. So it would probably bog down on your system.
DOOM?
This game can use your threads better, so you should do fairly well. You'd be at the BOTTOM of the chart, but that can get over 60FPS:
http://www.techspot.com/review/1173-doom-benchmarks/page5.html
Other games?
There are probably a lot of games that will run 1080p/60FPS or close to it. You'll just have to experiment.
Summary:
It varies a lot.
You can expect perhaps 60% average what a good i5/i7 would give with the same GPU and settings but again that completely depends on the games sampled.
TWEAK the game for the best results.