I've stumbled upon a character idea which has quite the potentially to screw someone if they run from you, and I want to know if it is legal in the rules.

A high elf rogue/cleric with the booming blade cantrip and war caster. In theory you get the damage from the weapon, sneak attack (if not used this round), divine strike (if you have it) and booming blade once you hit and the moment he moves...

As example, a rogue 3/cleric 8 character, with rapier would get
7d8 (1 rapier, 2 booming blade, 1 divine strike and 3 once he moves) +
2d6 (sneak attack)
+ dex bonus.

Is everything correct here per the rules?

If so, I think that's pretty insane for something that doesn't really costs any resources.

You usually get everything but Divine Strike (which only works during your turn)

• The rapier deals 1d8 + Dexterity modifier damage.


• 
  Booming blade as an 11th level character deals an extra 2d8 if the character doesn't move (3d8 more if they do).


• Sneak attack adds 2d6 even if you've used in a previous turn this round (see below, emphasis mine, and this related question).

Once per turn [not once per round], you can deal an extra 1d6 damage to one creature you hit with an attack if you have advantage on the attack roll. The attack must use a finesse or a ranged weapon.

The only thing you may not be able to apply is Divine Strike which says:

Once on each of the cleric’s turns when he or she hits a creature with a weapon attack, the cleric can cause the attack to deal an extra 1d8 ... damage to the target.

As such, you can only apply Divine Strike if the enemy is moving away during your turn.

The enemy has to move away for this to work

In order to deal all of this damage, the enemy does have to move away without using the Disengage action (see below). That is not necessarily an common thing to occur.

If you take the Disengage action, your movement doesn’t provoke opportunity attacks for the rest of the turn.

There is a great synergy, however, with dissonant whispers:

The target must make a Wisdom saving throw. On a failed save, it takes 3d6 psychic damage and must immediately use its reaction, if available, to move as far as its speed allows away from you.

This prompts the opportunity attack against the will of the enemy.

The opponent can stop moving at any time

If the enemy willingly provokes an opportunity attack (i.e. isn't moving because of dissonant whispers), and then you hit with booming blade, they can simply stop moving to prevent the extra 3d8 damage. This would be a very sensible thing for an enemy to do if they've witnessed you use the tactic before or if your table uses the Identifying a Spell rules in Xanathar's Guide to Everything:

To do so, a character can use their reaction to identify a spell as it’s being cast, or they can use an action on their turn to identify a spell by its effect after it is cast.

If the character perceived the casting, the spell’s effect, or both, the character can make an Intelligence (Arcana) check with the reaction or action. The DC equals 15 + the spell’s level. If the spell is cast as a class spell and the character is a member of that class, the check is made with advantage.

Applying Sneak Attack

Don't forget that applying Sneak Attack has it's own requirements:

...if you have advantage on the attack roll. The attack must use a finesse or a ranged weapon.

You don’t need advantage on the attack roll if another enemy of the target is within 5 feet of it, that enemy isn’t incapacitated, and you don’t have disadvantage on the attack roll.

While they aren't the hardest requirements to meet, they do limit the application of the damage to some extent.

You could get most of it

Unfortunately Divine Strike requires it to be your turn:

Once on each of your turns
when you hit a creature with a weapon attack. you can
cause the attack to deal an extra 1d8 radiant damage to
the target.

Which means for it to apply to an opportunity attack, that attack needs to happen on your turn, while not impossible, does limit it to the one instance you would get for your turn anyway (both Divine Smite and Sneak Attack).

So you could get that damage, but is dependent of the creature continuing to run after being hit by booming blade, and on Sneak Attack being applicable (either advantage or by another enemy being within 5 ft. of your target).

As a side note, Sneak Attack will apply on another turn even if you have used it on attack during yours as it condition is:

Once per turn, you can deal an extra [...]

No requirement of it being your turn.

This isn't especially unusual

Booming Blade combined with the War Caster feat is itself already known to be a relatively powerful combo because of how much it allows a character to take full advantage of the features of the spell. So your total damage is

• Damage which may benefit from a critical hit
  
  
  • 1d8 base weapon damage
  
  
  • +2d8 damage from Booming Blade's primary effect
  
  
  • +2d6 Sneak Attack damage
  
  


• Damage which may NOT benefit from a critical hit
  
  
  • +DEX Ability Modifier damage
  
  
  • +3d8 damage from Booming Blade's secondary effect
  
  


• +0d8 from Divine Strike, because Divine Strike requires it to be your turn (you may add it if someone triggers your reaction during your turn!)

This is the average DPA (Damage per Attack) for this character's Warcaster Opportunity Attack, assuming a Dexterity of 20.

beginarrayl
textName & textAC 0 & textAC 10 & textAC 13 & textAC 16 & textAC 18 & textAC 20 & textAC 25\ hline
textRogue3/Cleric8 & 38.075 & 38.075 & 34.175 & 28.325 & 24.425 & 20.525 & 10.775\
endarray

_{As always, harass me in chat if you need a breakdown on the methodology here}

Now, for the sake of comparison, let's try a different build: a Level 11 Arcane Trickster, having taken the Warcaster feat, who is wielding a Shadow Blade cast from a level 2 Spell Slot.

You weave together threads of shadow to create a sword of solidified gloom in your hand. This magic sword lasts until the spell ends. It counts as a simple melee weapon with which you are proficient. It deals 2d8 psychic damage on a hit and has the finesse, light, and thrown properties (range 20/60). In addition, when you use the sword to attack a target that is in dim light or darkness, you make the attack roll with advantage.

—Shadow Blade, Xanathar's Guide to Everything, pg. 164

With this, they have

• Damage which may benefit from a critical hit
  
  
  • 2d8 base weapon damage
  
  
  • +2d8 damage from Booming Blade's primary effect
  
  
  • +6d6 Sneak Attack damage
  
  


• Damage which may NOT benefit from a critical hit
  
  
  • +DEX Ability Modifier damage
  
  
  • +3d8 damage from Booming Blade's secondary effect
  
  

All of which gives them an average DPA of

beginarrayl
textName & textAC 0 & textAC 10 & textAC 13 & textAC 16 & textAC 18 & textAC 20 & textAC 25\ hline
textArcane Trickster 11 & 56.575 & 56.575 & 50.825 & 42.200 & 36.450 & 30.700 & 16.325\
endarray

This is, across the board, roughly a 50% increase in total DPA.

Not only is this character already dealing more damage on their Opportunity Attack, they're also a considerably simpler build, needing only a single class, and they no longer need to be a Variant Human to have the Warcaster feat, since they can just use the Rogue's extra ASI at level 10. This build also has the benefit that they can gain Advantage on their attacks in Dim Light (as an extra feature of the Shadow Blade spell).

Even if we ignore the Shadow Blade (maybe you're in Adventurer's League and your +1 is SCAG?), the damage is still better:

beginarrayl
textName & textAC 0 & textAC 10 & textAC 13 & textAC 16 & textAC 18 & textAC 20 & textAC 25\ hline
textAT 11 (No Shadow Blade) & 52.075 & 52.075 & 46.775 & 38.825 & 33.525 & 28.225 & 14.975\
endarray

The point is quite clear: your build is definitely powerful, but it's not outrageous or even uncommon for a level 11 character.