Gives the user a 20% chance to go first. This only makes the Pokemon go first if it is using a move of the same priority as the other Pokemon. Higher priority moves will still always go before lower priority moves, regardless of Quick Claw.
Deals damage and has a priority of -1. Ignores all accuracy and evasion modifiers.
Lets say two pokemon of different speeds were battling. The one with higher speed used vital throw and the Quick claw activated, and the one with the lower speed used synthesis.
The pokemon using synthesis would go first.(Tested on PO)
So,if quick claw activated, vital throw would go before all -1 priority moves, but still would go after normal priority moves.
See this for additional info