Q:

Is there any way to increase chances of a finishing move without commands or mods?

Is there is a way to increase the chances without mods or commands?

A:

Finishing / Kill 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. 
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. Eg, 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)
The only way to modify the in-game variables, 'KillMoveRandom', and 'DecapitationChance'  is by using console commands or mods, which you said you can't or won't be using.
That leaves you to having to resort to workarounds that will take advantage of the other factors listed above to improve your chances of triggering a finishing move -  i.e. try going for stealth kills, make sure both you and your enemy are standing on even terrain, try to get the enemy out of obstacles that may interrupt the finishing move animation, don't use the Slow Time dragon shout.

