Your wiring is simpler than what is depicted in the video, but the principles are all the same.
I would note that he tests continuity on the backs of the pots. That should be fine, but I generally do it on the ring of the jack. It shouldn't matter too much, but I do it at the jack because that's where the ground ultimately has to be. Once it is all installed, clip one lead to the strings and touch the ring of the jack to make sure they are grounded as well. You can do the same with the shielding if you add some.
I use a template like the one the guy uses in the video...all that means is that everything is arranged on a piece of thin wood or cardboard that has been drilled to be exactly like where the controls will go. That's important on my project, which is an ES. You already have a built in template, since your switch and pots all fit on a Tele control plate.
Did your kit come pre-wired? I have not seen one of these kits. The pics on the website don't show the back of the control plate. If it's prewired, then great! If not you'll need a wiring diagram. Caution, do not use the one on the Pitbull website because it's incomplete. Additional caution, there are at least two common types of Tele switch and they are wired a little differently. A pic of the back of your control plate would clear this up (for me, anyway...others may already know) ;-)
You can test the pups as in these videos. Your pickups should only have two leads, so you probably don't need to see much more than half the video ;-)