Overall, not too bad at all!!
You're pattern itself is good. You are making straight lines from barrel to barrel, and finishing each turn.
I like that you trotted him in a circle to start him off on the correct lead. When training one for barrels, it's important to help the horse get on the correct lead. When they are "finished" they will know what to do on their own.
The video is so bouncy, I can't really tell when he switched his lead for the 2nd barrel. Typically, when training a horse for barrels, once I finish the first barrel turn I will break the horse down to a trot, and ask them for the other lead. Of course, I've spent time ahead of time teaching the horse simple lead changes. Your horse seems to be doing pretty good so at this point, I would probably just let him figure it out rather than dropping down to a trot and switching him.
One thing to be careful of is that you do not let him turn the barrel too soon, and that you keep him on his hindquarters. Especially your third barrel, watch how his hind end swings out around himself. He's not planting and using it there.
I would definately use two hands at this point. It gives you better control and helps your horse. To "hold him off" the barrel and make him wait to turn, you should be lifting his inside shoulder with your inside rein (think of twisting and lifting your inside hand toward the direction of your outside shoulder) while also using your inside leg to ask him to bend around your leg. That will keep him shaped for the turn, but hold him off until the pocket is right to turn.
Keeping this shape throughout the turn will also help keep him on his hindquarters. Make sure you keep your weight even in the saddle, or even slightly back, to encourage to keep him weight back on his hindquarters.
Your first barrel was acutally your best barrel. Is there a reason you are starting on the left barrel? I think you should actually switch your pattern, and go to the right barrel first.
If you want any more tips, you can check out the thread we started for newbie barrel racers. Barrel Racing Exercises and Drills.