If you're using a Mac, suggest getting a player that works on top of iTunes such as Pure Music or Audiovarna for better sound quality. You can go to their respective websites to download trials. You can still use the free Remote app for your i-devices to control iTunes w/ these players on top.
If you're using a PC / Windows, then J River Media Center 20 will offer support for more formats (e.g. FLAC, DSD, etc) than iTunes. You can control JRMC w/ a tablet buying the Jremote program ($10). I don't think JRMC is that expensive either & you can try before you buy.
There is JRMC 20 for Windows as well & it's worth trying as an alternative to iTunes. It's very stable and responsive and will sound better than iTunes, but not sure it sounds better than Pure Music or Audiovarna. If you like how it sounds though, it's ultimately a better alternative than iTunes.
I wouldn't use an Apple TV as a go between - either hook the DAC directly to the computer using USB, or if you must go wireless, use an Airport Express (via minitoslink optical) that is bit perfect (Apple TV up samples to 48hz and audibly messes up the sound).
I have to say though that if you don't already have a large pre-existing music collection and plan or want to listen to a lot of material new or old, then going for a CD quality service like Tidal or Deezer makes a lot more sense. If I didn't already have a very large music collection, that's what I'd do.
