I've been through Christopher's lessons too, and watched the exact one that you're talking about. This problem comes up all the time, so know that you are not alone, everyone learning struggles with the F major chord. However, I have to point out that in the lessons, he does give you the answer... I can't remember which lesson it is, but he tells you to isolate the fingers trying to make the chord, and just focus on getting the notes to play cleanly.

I think this point gets missed by a lot of beginners... so, the tip/trick you're looking for is this:

1) Understand the chords (know in your head where your fingers are supposed to go).

2) Fret the entire chord.

3) Pluck each individual string slowly, listening for each to sound cleanly.

-all the strings sound good. Congrats! You're good to go on that chord
-you are getting some horrible twangs and buzzes. Failure... clearly, guitar is not your instrument as nearly everyone gets it on the first try, OPPS! I mean:

4) Isolate the offending note/string. Do this by removing all your fingers (from the fretboard, not your hand) and then place one finger at a time. Start with the first finger, fret with it, play the string/strings that finger is meant to be fretting. If it sounds good, add another finger and check again. If you encounter problems, take the last finger added away, check that all sounds good, then add the finger back and LOOK really carefully to see what you're doing wrong.

This seems laborious and you're thinking "man, I be Steve Vai didn't learn chords this way..." but he did. Everyone does. The problem is when you try to add all your fingers at once, and you have multiple issues to sort out. So break it down, be patient... it will come. Eventually you won't even have to think about it. Practice in little spurts, don't spend 15 minutes just trying to get the one chord. When you feel yourself getting frustrated, stop, put the guitar down for a bit, or practice something else.

-Last bit of advice: Avoid the "good enough" temptation... in other words, having buzzes and twangs, but what the hell, its good enough, right? Wrong. If you move forward playing the chord like that, it will become very difficult to correct later.

Stick with it!