Tried and true: you cannot open more than 1 DK char per server without buying another account, paying twice as much monthly fees and such.
BUT! You can create another DK on another server and transfer it anywhere you want. There's no prevention to transfer a DK to a server on which you already have a DK!
Don't believe me? Test it yourself with one of those free-transfer servers. Start a new DK on a high-pop server and transfer it for free to your low-pop server, on which there's already 1 DK. See for yourself.
What's it good for?
1. DKs make good bank alts. They've mounts and more bag slots than a lvl 1. Their unholy aura allows swift movement from the bank to the AH and the mailbox.
2. Some RPers like to have multiple DKs sometimes, of different races.
3. Professions. You can max most professions at level 65. Could be nice to have an extra DK around just to do your transmutes, potions or to make glyphs, and leveling from 55 to 65 is a lot easier than leveling a full character 1-80.
4. Probably more stuff I can't think of.