Now, I've been on a few playthroughs and never once have I ever backstabbed an opponent. Why? The first time I tried [Commander Maro at the docks during To Kill an Empire, if that makes any difference] I never succeeded. I was sneaking, hidden, with Mehrunes Razor. After many tries I just cut his head off. Now that I'm on my first thief/stealth character I want to start backstabbing people.

Am I doing something wrong?

The chances of doing the animation for backstabbing is completely random but seems to increase with the proficiency of your sneak. I personally load up on items with sneak enchantments before hand just to be safe

If you are using a dagger you have to get in close. Getting the distance right can take some practice, too close and the subject might hear you, not close enough and you are just stabbing air. Even with the silent run perk, its best to move slowly on the approach as this makes judging the distance easier and minimises the chance of detection.

As for the animation, sometimes I get it more often sometimes I don't. Why this should be I have no idea

You cut his head off? That has happen to me only with battle axes. For me stealth finishers with daggers have always been only the one where you incise a throat of enemy.

I encountered two backstabbing finishing animations. The necessary condition for these to be shown is that the total damage dealt in a single attack is bigger than the vitality of the opponent.

One animation is for daggers and swords and shows the player cutting the throat of the opponent. The other is for swords only and shows the player impaling the opponent with a sword while holding it from a shoulder

Backstabbing animations also happen with creatures not considered NPCs but still considered humanoids: I successfully backstabbed several Briarhearts

" The necessary condition for these [back-stabbing animations] to be shown is that the total damage dealt in a single attack is bigger than the vitality of the opponent"

Are you sure? I get a lot of one stab kills (i.e. the subject goes from full health to dead in one blow) but I get the animation maybe one time in five.

The throat-slitting animation plays when a main-hand sneak attack does enough damage kill the victim. So as soon as you get the 6x / 15x multipliers, it will play very often, I also managed several times to do one sneak attack for 50% damage, and stay undetected and do a second sneak attack and then get the animation.

Dual wield sweeps using daggers and the DB gloves will insta-kill almost everything, but no animation will play, instead the corpse is sent flying trough the air as if hit by a truck (well... 12 * 2 * 2 * 15 = 720 damage...)

I took out a greatsword.

I've gotten a couple of those throat slitting from behind animations with 1H swords. I don't use daggers as of yet.

So greatswords can do it also? I have played a greatsword user to lvl 30 and never cut any heads off with him.

No I mean I got angry at failed attempts so I just ran up and decapitated him.

I find the animation doesn't happen when you are too close.

Ok, I understand. Anyway I today first time saw it to happen with a greatsword: Beautiful swing on the neck of enemy and his head was cut off.

Using only a single main-hand dagger, I have gotten 3 distinct backstab animations:
1) 3rd Person camera: Reach up, grab target by the face, slit throat of target.
2) 1st Person camera: Reach up, hold target by the hair, scalp the target.
3) 3rd Person camera: Reach up, place target in choke-hold, plunge dagger into lower back of target.

As the other posters have mentioned, increased Sneak skill and getting the distance just right play the biggest parts in it. I see the animations on almost every kill.

Not seen it personally but a perk in the Two-Handed tree mentions decapitation, which would presumably be that.

I've cut many heads off with One-Handed blades. Earlier (whilst dual-wielding One-Handeds), I cut a vampire's head off by both swords crossing her neck at once. Was pretty epic.
One-Handed also has decapitation somewhere in the tree.

That's been my experience too. In one sequence I took out three Forsworn in a 30 second span and got the throat slitting animation for all three using exactly what you wrote. My sneak was 90+ and I had nearly every perk on the Sneak tree.

As far as distance from target before executing the move: it always seems to work best when you take a step or two closer to the target after the pickpocket prompt appears.

Keep moving toward them when you hit the trigger to attack, it will almost always give an animation.

2-Handed Perk: Devastating Blow: Standing power attacks do 25% bonus damage with a chance to decapitate.
1-Handed Perk: Savage Strike: Standing power attacks do 25% bonus damage with a chance to decapitate.

I just started to see these cut-head-offs after I pick that perk today. After my first character I have normally saved perk-points for most important perks and have wondered with every character why their finishers are boring: fewer of them and no head cuts. I thought it was connected to your weapon but now I'm happy to know it isn't.