My apologies, I thought your "beast strike" title was merely flavorful. If you are talking about a feat out of a Dragon Mag, you should better indicate that, as most people won't have any clue that it is even a feat name.
Edit: but of course, I see you did state in the initial sentence you had questions about this feat.... I failed my spot check...
So others actually know what you are talking about:
BEAST STRIKE [GENERAL]
Your martial training makes full use of your natural weapons in unarmed combat.
Prerequisites: Base attack bonus +5, Improved Unarmed Strike, claw or slam attack.
Benefit: When you make an unarmed strike or grapple check to deal damage, you may add your claw or slam damage to your unarmed strike or grapple damage.
Special: A fighter may select Beast Strike as a fighter bonus feat.
Dragon Magazine #355
Assuming a creature with claws is making an unarmed attack , does it get to add Strength bonus twice (one for unarmed + strength and one for claw + strength) or only extra "pure" claw (d4,d6 whatever without strength bonus)damage?
Unclear. An argument could be made for double strength bonus, but I wouldn't expect many DM's to let it fly.
Does the unarmed attack counts as bludgeoning, piercing and slashing(creature with claws as it has been stated)?
Assuming that you only get 1x Strength bonus, and your unarmed and claws were both 1d6 base damage, I think the damage would be:
1d6+[Str Mod] Bludgeoning, plus 1d6 P/S
Does vile damage apply assuming the creature has the Vile Natural Attack feat(for its claws)?
Each time the character deals damage with his natural attack, he deals 1 additional point of vile damage.
Hmmm.... That's a tough one.
It could be argued that you didn't make a natural attack (rather, you made an unarmed strike, which gets to add your natural attack damage, but still isn't a natural attack). But you could argue it the other way, that you've dealt damage with your natural weapons, and when you do, you deal vile damage. Basically it's a DM call.