Although Everyone has told you no, there is actually a incredibly simple way to be both a vampire and a werewolf: Hircine's ring, found in the quest Ill met by moonlight. Finish the quest and you'll get the Hircine's ring, which allows Werewolf transformations without giving you the immunity to all diseases. Become a vampire, put on the ring, and Beast Form should appear in your powers menu (under magic). You select it like a shout and you transform into a werewolf, but you are still a vampire and can drink blood and all that.
Additional info: http://elderscrolls.wikia.com/wiki/Hircine%27s_Ring_(Skyrim)
Well, I cured my own lycanthropy. I noticed there is one vote on this so I am sure that there are people who are still interested in this. I will leave as much detail as possible in the hope of helping people who have this issue in the future.
First of all, yes, the process for curing is exactly the same as how you cure Kodlak. You just have a witch's head in your inventory and examine that blue flame where you fought Kodlak's beast, and the option pops up, something like "Toss a head in the fire and cure your lycanthropy forever?"
Now, this is about the 300th time I'd tried activating that thing, so there is just some serious bug going on here. I think it's probably related to the fact that I spammed that flame the first time I was there during Kodlak's quest.
Here is the circumstance under which it finally worked:
To get the "purity" quest, continue doing random "quests" for Farkas and his brother until they tell you that they want to cure themselves.
I did Farkas and even though I used the flame to cure Farkas, I never got an option to cure myself. Then I did the same thing with his brother. When you have this quest, activating the blue flame causes the beast to come out right away.
After this, I tried to activate it, but nothing happened. Then I spoke to Vilkas and he said something like "Thanks, now I'm going to stay here for a while." After that, I tried the flame again and it worked.
I don't know if it finally working was related to what my actions were or whether it was just some random chance, but know that there is hope for those who think they will be stuck as a werewolf forever. Good luck.
Best Answer
DLC Dawnguard Information
If you have the DLC Dawnguard you can flip back and forth between the too. From the Skyrim Wiki:
"In Dawnguard, if you are cured of Lycanthropy or if you become a vampire lord you can talk to Aela and she can make you a Werewolf again and it cancels out your vampirism vice versa. If you would like to be a Vampire Lord again, speak to Serana."
Skyrim without Dawnguard
Without the DLC once you are cured of Lycanthropy. You can not get it again, with the exception Hircine's Ring. As long as you don't have Lycanthropy, you can contract Sanguinare Vampiris (which turns you into a vampire) from a vampire. There is a 10% chance to contract it when they cast Vampric Drain on you. If you cure it you can contract it again with the same method.
The important thing to note is Lycanthropy cures Sanguinare Vampiris. So if you have Sanguinare Vampiris and then get Lycanthropy you will cease being a vampire.