They are just plant material, I toss them on the ground (away from camp) or if I was going to be there a while dig a small hole to bury them in just so there's not a big pile of grounds laying around being ugly.
I have a Sea to Summit coffee dripper that works pretty well, I'm a pourover junkie, I do use one of those Melita paper filters with it though, I prefer the taste of a paper filter vs a metal filter but it works just fine with the metal filter than comes with it.
I do have a Hario manual grinder for when I'm car camping that I will use because whole beans are the way to go if you can spare the extra weight and space.
As far as the coffee itself I roast my own because I'm a hipster like that but I also do like to buy from small roasters. Three Keys Coffee out of Houston has an incredible Ethiopian coffee. Even if you don't buy hipster coffee any whole bean coffee ground right before brewing is a whole 'nother game from preground!