Vayne has a weak early game. This is balanced mainly due to the fact that she is pretty much the hardest late game ADC in League. Play with a support who can either heal you during trades, or who can lock down the enemy carry (read: Leona/Sona/Soraka).
Boots and 3 pots doesn't give her enough damage to be competitive in lane early, so Doran's Blade or Long Sword with 2 pots is a good choice.
Try to only engage when you KNOW you can outdamage the enemy carry, an equal trade isn't effective in a vayne lane, due to her low max health. Farming is ESSENTIAL. Her power relies mainly on getting early attack speed, so don't get the usual B.F. Sword unless you are far ahead.
This is where Vayne starts to pull ahead. Good engages will lead towards Vayne's victory. You should have a significant amount of lifesteal and burst. If you are behind, just keep farming, late game will come eventually.