Rear wheel is loose and wobbles at even the tightest setting I recently had my rear wheel and hub replaced because the older one was bent. When I came back home I found that the guy who fixed it left it loose, so much so that I could pick up the wheel and shake it about in its place. Also, it may seem that my rear cassette is a bit loose but the wheel was the only thing i could shake by hand 
                
            
I don't want to take the bike back to him, how can I fix this myself?
It's hard to see from the video exactly what is loose. However, you are presumably tightening the quick release properly so any looseness in the wheel must be from the bearings in the hub.
I see you have a 7 speed 'cassette'. It's quite likely that you actually have a 7 speed freewheel and relatively simple cup and cone style hub bearings. If that is the case you can attempt bearing play adjustment yourself.
In cup and cone hubs the axle is threaded, ball bearings are held between 'cups' in the hub and 'cones' threaded on the axle. Bearing play is adjusted by tightening of loosening the cones.
Here's an RJ the Bike Guy video where he's adjusting bearings after replacing a bent axle. It's a front wheel but it should give you and idea of what is involved. You can search for other videos or guides of course.
You'll need some special thin wrenches to adjust the cones and locknuts.