There are some tutorials for youtubers how to record audio. While it is for other reaons, the tipps there apply to your setup.
Basically, the entry level is a lavalier in the 50 bucks range.
You could try to put a cloth over your phone to reduce ambient breathing noises, similar those fluffy things are put over lavaliers and just try it out. The microphone in phones is not really optimised for all this, but tech got better, so maybe it will be enough for your tests.
Other than this, just use your gaming head set, if you have a good one.