How to know where a claw key can be used?

Solution 1:

Taken from Dragon Claws:

//Claw - Dungeon
Coral Dragon Claw - Yngol's Barrow
Diamond Claw - Skuldafn
Ebony Claw - Korvanjund
Emerald Dragon Claw - Reachwater Rock
Glass Claw - Forelhost
Golden Claw - Bleak Falls Barrow
Iron Claw - Valthume
Ivory Dragon Claw - Folgunthur, Reachwater Rock
Ruby Dragon Claw - Dead Men's Respite
Sapphire Dragon Claw - Shroud Hearth Barrow

EDIT: I just read your in game clause. I don't think there is a way in game to do this, but as a good role player ;), you should have a notepad at your side to make annotations when you are playing this game. BTW, I'm lazy, and when I don't remember something that my character should know, I just use gaming.se or google.

Solution 2:

Most dragon claws are really simple, use them in the dungeon you got them from (see the other answer and its link to UESP). They are usually a big deal, so just remember where you picked them up.

There is one dungeon where you need two claws: Reachwater Rock. You get one of the claws from another dungeon. When you have the quest to go there, you already have the other claw.

There are two gates/doors in Reachwater Rock. First, use the newly collected Emerald Dragon Claw. Then, use the Ivory Dragon Claw you got from an earlier dungeon in that quest.