The greatest online multiplayer arena-style FPS game of all time is...the original Unreal Tournament (1999) (aka UT99) which people still play online, greedily, to this day. Yes, it's 10 years old, but it has an amazing chemistry and feel that no other game has replicated. This game was so beloved back in the day that over 10,000 custom maps and mods were released for this game.
It has an excellent Deathmatch game (including Team Deathmatch or TDM) and also the best Capture-the-Flag game. There are still populated regular weapons custom map CTF servers and people still play 5v5 clan-match-style pickup-games (PUG) matches.
I recommend getting the Unreal Anthlogy which will give you the original Unreal single player game that preceeded Unreal Tournament, Unreal 2 (which isn't all that great), most importantly, Unreal Tournament (1999) itself, and UT 2004 which is a good game in its own right.
As for UT 2004, it is a great game in its own right but for different reasons. It's on-foot fragging games--DM and CTF--are inferior to those of UT99, but it shines for its vehicular game, Onslaught, which is great fun. There's also a populare user-made mod game called Invasion Monster Mash RPG which is essentially a non-competitive shooting gallery game where servers keep track of your player name and IP so that you can play a character and level it up over time.
Anyway, if you're look for a good FPS with a Deathmatch game, you definitely must consider UT99.
Link to Unreal Anthology for $10 shipped at Newegg.com
You might also be able to purchase and download the games over Impulse.
What about UT3? It's a sad story. Epic Games succumbed to consolitus and also hopped into bed with Midway, a dying game company. The end result was that UT3 ended up being a consolized abomination that was released months before it was ready and that really doesn't deserve the Unreal Tournament name. The feel of the game and the game play isn't too bad and it's worth looking at if you can buy it for under $10. It's not as bad now that it's been patched up, and is probably better than most FPS games on the market. It has low online player counts and overall it's just not as much fun as the original UT99 and the UT3 Warfare game isn't as much fun as UT 2004's Onslaught (which people still play online). The Deathmatch and CTF games aren't bad but personally I prefer the original UT99 for that.