Cheapest graphics card that has a better performance than current gen consoles?

Solution
An r7 360 (which is generally cheaper than the 750ti as well as being more powerful) is your cheapest option on this front.
A 370 would be preferable at the right price depending on where you are.
I'd say a 950 too but here in the UK the 370 is substantially (25%) cheaper
As stavrosmast said. The 750Ti would be the best choice. I currently have a 750 Ti FTW edition myself and I have to say that for a cheaper end GPU it performs pretty damn good with the right settings.
 
Theoretically speaking console-like would be something along the lines of GTX 650/Radeon HD 7850, but since on the PC you've got time on your side (or at least when you're a potential buyer at a given time with no past) you could go for a GTX 950/Radeon R7 370 and be pretty sure the performance not only matches the likes of PS4/X1 but beats them as well.

P.S. In the same price-range, alternatively you could pick up a refurbished GTX 960, which is more than significantly superior when compared to PS4/X1.
 


Same thing applies to ordinary used cards. As a matter of fact, it's more likely for those to malfunction as they're not double-checked for issues, and in case they do, you'll usually have to throw 'em in the trash can.
 
An r7 360 (which is generally cheaper than the 750ti as well as being more powerful) is your cheapest option on this front.
A 370 would be preferable at the right price depending on where you are.
I'd say a 950 too but here in the UK the 370 is substantially (25%) cheaper
 
Solution