I've been searching, from what I can find, it's nearly impossible to get a pre-built computer for under $800 with a video card with 4GB of VRAM. All of the other specifications are easy to achieve for less.
If it consoles you, the difference between 2GB and 4GB of VRAM in low end cards is negligible to nonexistent. Therefore, the pre-built computer I recommend to you contains a 2GB GT 730.
Dell Inspiron 3650 - US - UK
- Price: $619.99 USA, £439 UK
- CPU: i5-6400, 2.7Ghz base, 3.3Ghz turbo.
- GPU: GT 730, 2GB VRAM
- Disk: Generic 1TB 7200RPM HDD
- RAM: Generic 1x8GB DDR3L-1600Mhz
- WiFi: 802.11 b/g/n
Dell Inspiron 3650
If you were thinking - "oh the high end graphics cards have 4GB of VRAM, so 4GB of VRAM automatically classifies a GPU as good," you were completely and totally wrong. In fact, if you were to custom build a computer, a 2GB GTX 960 would be vastly superior to a 4GB GT 730, and the performance difference between the 4GB GT 730 and the 2GB GT 730 would be practically nonexistent.
In case you still doubt this, here is a link to a 3DMark Firestrike benchmark comparison
Both graphics cards have 2GB VRAM.
Furthermore, clock speed also doesn't dictate performance. Intel has a vastly superior IPC (Instructions per cycle) than AMD, and therefore can provide incredibly higher levels of performance with a 2.7Ghz clock speed than AMD can with 3Ghz or even 4Ghz.
2.7Ghz i5-6400, quad core: 521
3.5Ghz FX-6300, hexa core: 412
You can see how clock speed, and even amount of cores bow down before IPC in the benchmark above.
Therefore, I think that this will have plenty of power, and I have proved that even if there were a way to make the GPU have more VRAM and the CPU more clock speed, it would not matter.
Hopefully you take these benchmarks and tips into consideration when purchasing.
If this does not suit your needs, please notify me. If there is a particular reason you need exactly 3Ghz clock speed or 4GB VRAM, also please notify me.