Since 5th edition is about eight years old, I'm guessing everyone has its own version of the Champion by now. Here's mine.
The idea is not doing a complete overhaul; just the minimum changes necessary to make it closer to the Battlemaster in damage output (and give it some out of combat utility), without adding much complexity, so the the Champion remains the "simple fighter".
I think a couple of small changes at levels 3 and 7 might be enough. This is because Improved Critical and Remarkable Athlete are especially weak; if we fix them early on, we fix the entire subclass.
Of course if you already like the champion as written, you can use the original version. As always, I advise making all house rules optional.
If you like this, I might do the barbarian (berserker) and ranger (hunter) next. Currently, I'm only interested in the SRD... The beastmaster ranger would take more rewriting. I'm not sure the other SRD classes need additional boosts; maybe the assassin. The warlock an the sorcerer I find a bit lackluster and redundant with wizard, but, well, they are good enough I guess. Paladin, wizard, cleric, druid, rogue, are all more or less balanced IMO (if anything, maybe fighters and barbariasn need a bigger boost to compete with them).
So, now the Champion can be awesome at grappling (or at least escaping from a grapple) and deal decent damage throughout his career.
The boost is STILL not enough to outshine the Battlemaster, but it makes both Improved Critical and Remarkable Athlete good features.
Giving the entire proficiency bonus to damage at level 3 would work too, although I'm a bit concerned that in this case everyone would take three levels in fighter just for that (and action surge, etc.). There might be some other balance issue here (maybe with multi-classing), but I don't see it.