What determines when you do a finishing move?

The official strategy guide confirms that they are random, and that they do no extra damage: they're pure aesthetics. It suggests that you can increase your chances by executing a power attack on the last enemy in a group.

If you are within about 1-4 blows of killing an opponent, a "finishing move" animation may appear, depicting one or several fatal strikes from either a first-person view or third-person "KillCam". If the Savage Strike (One-handed) or Devastating Blow (Two-handed) perks are unlocked, there's a chance that this will be a decapitation. Note that under the right circumstances, NPCs and Dragons will also be able to perform finishing moves, both against each other and against you.

^{Source: UESP wiki, "Skyrim: Combat" article}
Finishing / Kill moves are not completely random. Finishing moves are affected by the in-game variables, 'KillMoveRandom' (default: 50%), and 'DecapitationChance' (default: 40%) and various factors, some of which are:

• Kill Cameras will only be performed if killing the enemy will take the player out of the combat mode. Due to this, killing the last active/aggressive enemy will often trigger a finishing move / kill cam. But this does not mean that only killing the last active/aggressive enemy will trigger a kill cam, e.g. killing an enemy while sneaking and remaining undetected will allow a Kill Camera, even if there are more enemies present.
• The opponent being killed. Some enemies in the game seem to have no finishing moves associated with them.
• Your position relative to the enemy. For example, there is a one-handed sword stab-through-the-middle execution that will only play if you're standing behind a humanoid opponent. There are also finishing moves which require your enemy to have surrendered and fallen down to their hands and knees.
• It should be noted that your steath state has an affect as well. E.g., you can cut throats with a dagger.
• The terrain: finishing moves appear to depend on relative elevation. While it's possible to enter into a finishing move animation on uneven terrain, it's far more likely to happen if both you and the enemy are standing on a flat, level surface.
• Other nearby obstacles: If the enemy NPC is too close to an object like a table, it may interrupt the animation and leaving you standing there until enough time has passed and the game gives you back control of your character.
• Chance: Some animations seem to play more often than others.
• Slow Time: Finishing moves will not trigger while under the effects of the dragon shout Slow Time.

^{Source: UESP wiki, "Skyrim: Combat" talk page (edited to take into account the comments in reply to the original post) | TES wiki, "Kill Camera" article (edited to explain further)}
You can modify the finishing move-related variables using the console commands, set killmoverandom to <0-100> and set decapitationchance to <0-100>. Or, use mods like SkyTweak (configuration page screenshot), The Dance of Death - A Killmove Mod (configuration page screenshot) or Killmove Control which have UIs to help configure those settings. 'The Dance of Death' mod can even remove the 'last active enemy' restriction.

Well, based on my experience (255 Hours of Gameplay) Skyrim those above, seem to be true for me, with every character.

Stealth Assassination Finishing Moves

• Happens most of the time when the target is a human and is standing up. Any direction will work here, from sides, behind or even in the belly.
• Will only happen if you have enough damage to one hit kill the target on your sneak atack. If your damage is not able to one shot the enemy, no animation will be shown, just a normal bonus sneak atack. As it is a finishing move, the enemy won't stay alive after the hit.

Close Ranged Finishing Moves ( Both One handed and Two Handed Weapons )

• Holding the sprint button and running towards a enemy from a close range (Not running the entire map, it has to be close) and power atacking happens to proc the animation more offen. Even without perks in those trees.

• Again, the damage must be high enough to kill the enemy. As it is a finishing move, the enemy won't stay alive after the hit.

More Skills or Critical Strike Chance Mean More Finishing Moves:

Since the damage must be high enough to kill the target for the finishing move to happen, if you have a higher one handed skill level for instance, your damage output will be higher, allowing you to execute enemies more frequently.

Same applies for critical, since critical helps the damage output, with a higher damage, you will be able to kill enemies that you wouldn't be able to without a critical strike.

Needs Testing Here:

I am not sure if the enchantings and potions bonuses apply after or before the sneak atack multiplier. But if the damage calculated for the multiplier involves the extra damage from enchantings and potions, it would also help you to get more animations.