To perform more than one attack/round, it usually requires a full-attack action.
As natural attacks, you don't receive additional attacks from your attack bonus (see
here). Instead you're able to do up to 2 attacks with the claws and there's no penalty for 2-Wpn Fighting or stuff like that, because it's a natural weapon that grants 2 attacks (one for each claw).
If you had another kind of natural weapon (ex.: if you had a tail, or a pair of horns) and you wanted to perform additional attacks with them, those would be considered secondary.
Now... you want to combine it with "unarmed strike"... I'll have to brush up on the combination of unarmed strikes and natural attacks and get back to you on that... brb
EDIT: Jackinthegreen got it right. Weapon/Unarmed strikes are performed first, and the natural weapon attacks go in as well as secondary weapons (-5 penalty, 1/2 STR) , but you would still be allowed 2 attacks with the natural weapons with the same penalty.