My display won't turn on even when computer does, any ideas?

If it's not working though the screen and the HDMI it sounds like the graphics card has been fried. I don't know how old your laptop is, but if's the one I just found via Google it's probably not economically viable to repair it (new main board).

You should be able to pull the HDD out though and get your data off it, it might even boot if you can find another laptop the same as yours.

I do not believe it is a port issue. First, try turning on your laptop and press "F8" to enter "Safe mode." If your system loads, click the "Start" button and type "device manager" in the search bar. While in device manager check to see if any drivers are malfunctioning. If this does not work, your operating system (Win7, Win8, or Win10) might be malfunctioning. If you are able to remove your hard drive, do so first to get your files from it. Afterwards, plug your hard drive back in and try reinstalling your operating system. You should have an operating system disk that came with your laptop. If not, you can purchase one online. Hopefully this can help you with your issue.