Team Deathmatch is the only deathmatch option. Look for it in your server browser.
I think single player was compelling from a story and action (Dima side story) but the gameplay just didn't hit on all cylinders for it. Battlefield 3, to me, is always about epic large sized multiplayer with vehicles blowing everything to shit. That's what Battlefield is to me. You'll discover that as you play more and improve.