Looking to start

Hey everyone, I was looking to start playing Vayne, so I'd need a nice build for her- the current one I have in mind is:

Bloodthirster Berserker's Greaves Infinity Edge/situational Ad item Blade of the Ruined King

Now, I was debating between these for the last 2:

Runaan's Hurricane- I feel it'd give me more teamfight potential Phantom dancer- Seems like it'd work well with Vayne's passive and moving through units is a nice + Statikk Shiv- More wave clear, Ap damage, and also has potential to be useful in teamfights Rapid Firecannon- Good poke, Ap damage, and since Vayne only has a range of 550 it would be a great way to suffice for her lacklustre range- also would be a good choice against Caitlyn(I think?) So this may be situational..?

I'd also like to know what order I should upgrade my abilities, I generally start with Silver Bolts unless I'd need the mobility of tumble

I'd appreciate any/all opinions on this, and possibly other helpful little tips or tricks if you're willing to share :)

Thanks! ^-^