If price is of concern, I would recommend looking at getting a high quality dac/amp or dac and amp separates with an interface. My general budget gaming recommendation for the recording side would be a Focusrite with a Shure SM57 or SM58 and a pop filter. I also highly reccomend the DBX 286s here if budget allows for a higher quality preamp and effects. You more than likely will want a dynamic mic for the background noise rejection. Time tested equipment that can produce strong results with some recording fundamentals and not too hard on the pocketbook. This would open up additional budget for your headphone gear. Almost every professional I know has a SM57/58 in their mic cabinet.
My go to recommendation at a budget is almost always the JDS Labs Atom/Atom Dac. Both are excellent pieces of kit and the service from JDS is top notch. Combine this with some software based eq and you should be able to get a great sound. The Monolith by monoprice would also be a nice option for more money if you are set on the combo unit. Most stuff on Amir's recommended tier will all also be excellent so this part will be flexible. I like over 100 milliwatts into 300 ohms to allow for some eq overhead on the HD800(S)
For overall performance with budget in mind, you will have better luck separating your dac/amp from your interface. Getting stuff piece wise also allows for more convenience swapping gear in the future should you change your mind about any of it.
In summary,
Budget Recommendation
Focusrite Scarlett 2i2 Gen 3
Shure SM57 (or 58) and a pop filter (If you prefer condenser the Rode NT1 is a strong offering for the price)
JDS Atom and Atom DAC(separates) or Monoprice Monilith(combo), or any other high performing kit with 100 milliwatts+ into 300 ohm
Software eq of your choosing a free recommendation would be Equalizer APO with the Peace GUI plugin
Optional but Recommended for Budget
DBX 286s mic pre and processor (has nice preamp, compressor, de-esser, etc)
Higher price Recommendation
RME ADI 2 Pro FS
Shure SM7B (SM57/58 could work here as well to save a few bucks)
DBX 286s mic pre and processor (has nice preamp, compressor, de-esser, etc)
I guess I like to type a lot. Hope this might provide a little clarification and a much more budget friendly option.