Infernal Barb Hellfire Claw + Monk Martial Arts?

Under the Path of the Infernal Barbarian, you get the "Hellfire Claw" ability, which states the following:

"If you don’t deal this fire damage with a weapon attack on your turn, you can use a bonus action to make an unarmed strike with the claw, provided it’s empty. On a hit, the target takes the normal damage from your unarmed strike, which can be either bludgeoning or slashing damage (your choice), plus an extra 1d6 fire damage."

How would this pair with the Monk's Martial arts, specifically the following:

"Bonus Unarmed Strike. You can make an Unarmed Strike as a Bonus Action.
Martial Arts Die. You can roll 1d6 in place of the normal damage of your Unarmed Strike or Monk weapons. This die changes as you gain Monk levels, as shown in the Martial Arts column of the Monk Features table."

I mean I assume with the Martial arts die would be pretty simple, you just add the hellfire claw dammage on top of the original damage, but the main question I have is for the Bonus Unarmed strike. Would that bonus strike be a part of the infernal barbarian strike? Another strike after it? Either way, I think this could be a really interesting build overall. Oh, and if you guys have any suggestion for my monk subclass/how to do leveling split for a level 12 character, I'd love to hear it!

Wrong information.

Quote from Motley711 >>

Under the Path of the Infernal Barbarian, you get the "Hellfire Claw" ability, which states the following:

"If you don’t deal this fire damage with a weapon attack on your turn, you can use a bonus action to make an unarmed strike with the claw, provided it’s empty. On a hit, the target takes the normal damage from your unarmed strike, which can be either bludgeoning or slashing damage (your choice), plus an extra 1d6 fire damage."

How would this pair with the Monk's Martial arts, specifically the following:

"Bonus Unarmed Strike. You can make an Unarmed Strike as a Bonus Action.
Martial Arts Die. You can roll 1d6 in place of the normal damage of your Unarmed Strike or Monk weapons. This die changes as you gain Monk levels, as shown in the Martial Arts column of the Monk Features table."

I mean I assume with the Martial arts die would be pretty simple, you just add the hellfire claw dammage on top of the original damage, but the main question I have is for the Bonus Unarmed strike. Would that bonus strike be a part of the infernal barbarian strike? Another strike after it? Either way, I think this could be a really interesting build overall. Oh, and if you guys have any suggestion for my monk subclass/how to do leveling split for a level 12 character, I'd love to hear it!

I would say you can add the extra 1d6 fire damage to your Martial Arts damage. Contrary to what Character77006 said, I do not believe that this use of the word "normal" specifically refers to the base (non-Monk) Unarmed Strike damage calculation; it seems to refer to just whatever your Unarmed Strike damage is without Hellfire Claw.

As for your other question, both features give you a way to make an Unarmed Strike with a bonus action. Since you only have one bonus action per turn, you can only do one of them on a turn, not both.