Dragon Claw is an awesome move for taking out dragon-type pokemon because it's one of the few moves in that family which is readily available to a lot of different pokemon types. Don't be surprised if a bruiser of a completely different type tears apart your pokemon with a surprise Dragon Claw. This makes it a great boon to any team, dragon-typed or not - so where's the TM?
Dragon Claw as a TM isn't available until after you beat the game a single time. This means travelling to the Pokemon League and beating the Elite Four and the champion at least once. After doing so, you'll open up a lot of new territory, including the path to Icirrus Town in the west of Unova. Head there once it's open and you'll find another new location: Dragonspiral Tower, a place that's rife with dragon-type pokemon.
Enter the tower and climb. After the first floor you'll stop running into pokemon, so you'll have little trouble wandering the rest of the way up. Once you get to the fifth level you'll find Dragon Claw on the floor. Snag it and you can leave...
... or can you? It's not advisable to enter Dragonspiral Tower unless you have encountered N on Victory Road. Follow the Zoroark that appears when you ascend from the Pokemon League into the mountainside. It will eventually lead you to N's old castle, where you can fight his legendary-type pokemon, depending on which version of the game you own. When you beat said pokemon you'll receive a special stone which can be converted back into the pokemon you fought at the top of Dragonspiral Tower. Do that first to economize your time.