Yes, you would be able to run it with your computer off. A VPS is essentially another computer accessed online, your able to use it as you would a normal computer only difference is that you would need to connect to it through VNC viewer or some sort of RDP Client.
The way it works is you purchase it, download the relevant software (VNC Viewer) on both your computer and the VPS (using terminal) and then you should be able to use it through VNC viewer and view it as a normal Desktop.
Hope that helps.