First off, welcome to WoW. Secondly - when it comes to PvP, there is a pretty steep learning curve... it's not just "knowing the mechanics". It's knowing the core abilities of EVERY CLASS in the game, and how they interact. As a death knight, you need to get experience with how different classes will try to take you on. A rogue will not attack you the same way a mage would, or a hunter, or a monk, et cetera. Practicing on dummies really won't be helpful; dummies don't hit back, dummies don't CC you, dummies don't move, etc.
If you're interested in PvP, just keep practicing. Random battlegrounds (NOT the same as rated battlegrounds) are a great place to start. Try to learn the objectives of each battleground - even if your gear/skill are not elite, you can still help your team win if you know what you're doing. As far as gear - until you get into the L85-L90 range, honor and conquest points are hard to come by. You can visit the auction house to see if there's any decent crafted PvP gear available - lower level crafted gear can sometimes be very abundant (and cheap) since people will make gobs of the stuff while they're levelling their tradeskills.
Best of luck. :-)
Welcome! There's some good advice in this thread.
Some realms have locations where players gather to duel, generally outside the entrance to capital cities. Bring plenty of bandages and the like.
It's worth noting that dueling isn't the same as organized PvP, largely because 1v1 is a whole different ball game. In a duel, using all your cooldown abilities is an option. In group PvP, you'll need to learn to make do with what you have in the moment (and some players will intentionally engage in duels while important abilities are on cooldown specifically to try to develop techniques to compensate for not having them). Nonetheless, I've personally found that dueling teaches very useful PvP skills that can be applied elsewhere and a whole lot about what each class and spec can be capable of.
If you're friendly and a good sport, you can make friends with players who are skilled at classes that give you trouble, and such players might be willing to help you figure out how you can do better against their class.
Finally, don't underestimate the power of arranging your control scheme to suit the way you want to play. Many players find it useful to learn how to use their mouse to move, and key-bind all their important abilities. If you haven't already done so, it might be worth looking into. Having critical abilities available at the split second you need them makes all the difference.