First of all, I really respect that you came here to ask for help and were willing to submit a video of your gameplay. This is the absolute BEST WAY to get advice, even though it doesn’t always feel like a welcome idea.
That being said, let’s break down the game:
0:15, insta soulburn gateway so you have a way to run and create distance; great job!
0:18-20. You’re in his LoS before you start casting charhound, which is the right cast, but you literally waste 2s while he’s rooted and healer is far away for 0 reason to start casting. No reason to hold “A” for those 2 seconds to strafe.
0:22 first cast of charhound = good. Second cast of shadowbolt is bad. Next cast should just be dogs into tyrant so you get the extension on dogs. Shadowbolt is pretty pointless here because you have shards to spend, it does very little damage, and you need to get other demons rolling over pressing shadowbolt.
0:24- you stun him for your damage, great job! Exactly what you wanna do here 
0:25 you start casting tyrant. good in theory, but you don’t have dogs out, so your tyrant is gonna be way weaker.
0:26-0:28, tyrant is cast and you spend 2 globals and 2s walking to the side for no reason without applying your curses or using an instant demonbolt proc, so you’re literally just CCing yourself for 2s. You started combat at 18s and are at 28s and literally 4 of those 10 seconds or 40% of your opener you’ve wasted by just walking to the side. If you planted, you’d already have dogs out and a hand of guldan if you just sat your butt still and casted and have 2x the damage available.
Also this DH fat fingers nether or something, so you swap to the pally with your felguard and that’s super good!
0:29-0:30 you use 2 damage globals in dog and your demonbolt proc; good job! Get that damage rolling! However, if you did things correctly, you’d have damage ramped and pally would be at 70% instead of 90.
0:31you use your aoe fear -which in 2s in general feels like a WAY worse option than coil-… and on the DH? Who has just used a major defensive AND enemy healer is out of CC? Straight to jail. This is the first real big mistake. Against any competent team, that healer dispels the fear instantly and you get your butt pushed in by a DH on fear dr with no peels for yourself because you just used both stuns AND your peel AND your healer’s root. It’s also important to note that the DH has pressed 0 offensive buttons and your healer hasn’t been CC’d yet, so you’re in a SUPER vulnerable position because you used everything offensively and didn’t really generate pressure.
0:32 this dh… trinkets and backflips??? What on earth?! HIS HEALER HAD DISPEL UP?!
0:33 you 1/2 fear his trinket. Again, not a good time to fear the DH. He has pressed 0 offensive buttons, so what are you trying to CC? These are YOUR CDS!!! You want to do damage to him and make him scared!!! If you feared the healer here, THAT would be huge.
0:34 you try to shadowfury off the fear, but that’s not really impactful because it’d be 1/2 anyways and he BACKFLIPS THE STUN OMG THIS GUY IS INSANE?!
0:34-36, goood you’re pressing damage again, PROUD OF YOU!!! This all being said, you have been in combat for 16 seconds and pressed only 5 damage globals; one of which was shadowbolt, which is about as useless as it gets. That’s ~11 seconds of missed damage and wasted CC, or 68.75% of your time in combat being effectively useless. I’m 36 seconds into a 213 video, 16 of which was combat, and I can already tell you’re not doing your pve rotation correctly, you don’t know when to CC, and we’re wasting globals.
That being said, you’re already 2x better than this DH at least and we’re here to learn! Let’s keep watching and figure out how we can improve!