Help me thicken higher range? (video of me attached)

Not sure what you're looking for.

Chris Cornell is a very "chesty" singer - his highs sound heady and full in part because of where he places his voice and in part because of the appropriate application of reverb.

Hard to tell which notes you're trying to improve. Could you give some guidance, like time stamps or something??
 
You're probably denying a register change.
For guy singers the natural tendency is to try to get louder to get higher. Thus we actually ended up straining and decreasing our volume.
We need to let our folds actually thin out (think falsetto) put "push" a little harder through the air we bring up.

Cornell is doing precisely that. Combining pushing more air while thinning his vocal folds and moving to his "head" voice.
 
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