Kadano recently published a large amount of data regarding input lag with Faster Melee and different Monitors/Graphic cards. You can find that
here.
To save you some time, the optimal setup that is
confirmed to work well is:
a decent CPU,
a Gtx 900 Series graphics card from Nvidia and
a powerful CRT Monitor set at 120hz.
The choice of CPU probably doesn't affect the input lag as long as it can run dolphin without stutters. Dolphin isn't very demanding overall but it requires decent single core performance because it can only take advantage of up to two cores and mainly of only one core.
Both Kadano and I recommend the G4560 processor from Intel because the processor has amazing performance for its cheap price and is powerful enough not to limit(bottleneck) graphics cards below 200$ in most games.
Regarding the GPU, it is possible to use the integrated graphics of the G4560 but they still cause minor amounts of lag at the lowest graphics settings. In order to use the integrated graphics with a Monitor that has more than 60hz, you will need to make sure that the motherboard has video connectors that support more than 60hz. The integrated graphics of the G4560 aren't able to output analog video(for a CRT) at more than 60hz, making it unsuitable to use with a CRT.
If you want the lowest amount of lag, 900 series graphics cards from Nvidia have shown not to introduce lag while also being able to allow high amounts of enhancements. In particular, the models GTX 950, 960 and 970 were tested, all of which are very good cards overall and will play most modern games. Older Models probably work fine too but haven't been tested yet.
It is important to note that the new 1000 Series graphics cards are not able to output analog video. This means, in order to use a CRT Monitor, a 900 Series or older graphics card is necessary.
As I said earlier, a powerful CRT Monitor is the best Monitor to get because they don't have any inherent input lag and are able to display in 120hz. You will want to get a powerful CRT Monitor, meaning a CRT Monitor with a high horizontal scan frequency, which is measured in kHz. The higher the horizontal scan frequency, the higher the resolution you can get at 120hz. A horizontal scan frequency of 120khz or higher is recommended. Setting up the monitor will take a bit of time as you will have to enter the resolutions and refresh rates you want to use manually. Finding the best resolution you can squeeze out of your monitor needs a bit of trial and error.
In general CRT Monitors can be bought very cheap (used of course) and have very pretty colors in my opinion, especially the dark/black colors are gorgeous. They also can be viewed from all angles without noticeable loss of image quality unlike LCDs. However, they tend to use a lot of power and are quite bulky and heavy. This is the best choice if you are willing to invest a bit of time and have the space for it.
The best alternative to a CRT Monitor is a 144hz Monitor. The Asus VG248QE was tested to have extremely low amounts of latency, even compared with other 144hz monitors [not by Kadano]. Last weekend I brought my own VG248QE to Kadano's place and we did some tests, you can find them in the table I linked above. Running the monitor at 120hz and using the program "Strobelight" at the 10% strobing setting, we were able to achieve results that were only about 3ms slower than with a CRT at 120hz. The reason why running the monitor at 144hz isn't optimal is because the image doesn't look very nice because the refresh rates of the game and the monitor don't match. I think it is a considerable substitute for a CRT. The Monitor is really nice for gaming in general since it will do 144hz at 1080p and the 16:9 aspect ratio is great for watching videos and movies. The colors unfortunately aren't very nice but they can be adjusted to look ok. If you play a lot of different games and you're willing to spend like 250$, I would get the VG248QE even though it has slightly higher input lag.
The other parts don't seem to make a significant impact on performance. If you want some suggestions, I made some recommendations
here (starting from "edit:").
