As one might guess, to complete a Battle Pass you need to go up in rank, and to do that, you need XP. Players are currently awarded 50 XP for completing a matchmade game, and more XP can be earned with weekly challenges. The good news; you'll get 50 XP no matter what you play in matchmaking, or how you play it. The bad news; it'll take a while to obtain this for free.
The easiest and quickest solution is to keep buying XP grants until you've gotten to rank 100 in the current Battle Pass.
The other is to patiently play the game and progress normally, but there are still ways to go a little faster. So here's a few tips to help out, although keep in mind that these are not "one size fits all" hints.
1. Play in playlists that go faster. So far they all seem to move at about the same pace, however more playlists will likely be added. For now, Big Team Battle seems to me to move the slowest, while Ranked seems to move faster. EDIT #2: With the addition of more playlists since the creation of this guide, other playlists like Tactical Slayer and Fiesta both move much quicker on average, due to the fast paced nature of both game types.
2. Don't leave games just because you're losing. This ends up just being a waste of time. Even if you lose, you'll still get your XP.
3. Focus on your weekly challenges. These have nice XP payouts if you can finish them. Some of them will be skill based, but in most cases those offer more XP. Completing a weekly challenge can mean saving you from having to play an extra 2-6 games.
4. Save any XP boosts you get for the start of each week. Once new weekly challenges are in, use an XP boost and focus up on finishing those challenges. You can easily rack in about 200-600 XP in one challenge with an XP boost active. That's 4-12 games you won't have to play now, from one challenge.
With zero XP grants and using the above tips, I've gone from Rank 1 (November 19th) to Rank 47 (November 30th.) I haven't been playing all day every day, I've just been getting on when I can, and finishing all my challenges with XP boosts active. Obviously there's no rush in getting this finished, since even when the new battle pass arrives, this one should still be here. But still, this should help save some time.
Happy hunting! If anyone has any extra helpful tips please comment and I'll add them into the solution (you'll get credit for any help, of course). If you feel the need to downvote, please explain what I can do to make the solution more clear.
EDIT #1: With a new update, 343 has implemented a new daily system. Players will now get 300 XP for their first game, and subsequent games will decrease after each one in extra XP (200 XP > 150 XP > 100 XP). I believe once you hit your seventh game of the day, it goes back to awarding 50 XP per game. That being said, I would recommend playing at least one game a day. This just released so I have yet to test if it resets daily as I have been told, but I'll come back and update the solution if it doesn't.
EDIT #2: I know a lot of people are saying to do Bot Bootcamp, but personally I felt it wasn't rewarding enough. For one thing, it is extraordinarily boring, and for another I found that I was actually having to deal with others idly boosting while I did all the work. A lot of weekly challenges can't even be completed in Bot Bootcamp, so I would recommend staying in normal playlists if you feel that what I've mentioned above isn't for you.