Last few days when I turn the ignition key everything lights up but the car doesn't start. It used to do that once in a while and from a second try it would usually start. But last few days it does it all the time. As you can see on the video I turn the key, everything lights up and nothing happens... Could that be a battery issue?

 

My guess its the battery or starter, more than likely starter. Check battery with multimeter, should be around 12V give or take a little with car off & about 14V with it on. If battery checks out its most likely the starter, try hitting the starter with something a few times & try starting the car again, sometimes that helps but would only be temporarily.

 

batt is super low, or the starter solenoid has taken a crap....

When this occur's the easiest way to check is to replace with a TESTED stronger battery with the proper Cold Cranking AMP's and if that doesn't work the next step is to replace the starter as the solenoid cannot easily be replaced (they usually come sold together)

Before buying a alternator I would check fuses just to be on the safe side.