The only way to spawn the enderdragon in vanilla without 1.7 is to just go to the end, but to do that you'll need to find it.
Note: if you're in survival, be sure to bring at least four soulsand and three wither skeleton heads.
If you're in survival, you'll need to gather enderpearls and blaze powder to get eyes of ender. Then you need to throw them, and follow them in that direction. (if you're in creative you can just get them out of your inventory)
Once the eyes of ender start falling, instead of going in some other direction, dig directly down where it fell. After some digging, you should find stone bricks, and directly below that will be a stronghold. In strongholds, there is usually a room designated to the ender portal, which has a silverfish spawner and lots of lava. Use (right click) on the portal frame (what's around the biggest plot of lava) with the ender eyes and when all of their buttons are up, a black portal will appear within the frame. Jump in!
There will be an enderdragon in the end. (where you just got teleported)
Use the heads you brought with you/search "wither skeleton" in your creative inventory to get wither skeleton heads, and use the soulsand with it, in this pattern: (h = wither skeleton head, s = soulsand)
After building this (not in a crafting table, in the world) a wither will spawn. Have fun!