I'm writing this under the assumption that you're new to the game and don't know how anything works (such as if you're doing one of those "play this app x amount and get money for it" deals). Apologies if this isn't what you need, or if this is obvious/unhelpful information.
You're almost certainly going to hit L+ with one of the legendary-tier (fodder) heroes first, since they're much much more common than ascended-tier heroes. You need at least eight of the hero itself, then a further 48 other fodder heroes in the same faction (in multiples of three) to branch your way up.
Sure enough, the fastest way is to just pay money and buy the resources to get fodder heroes. But if you're trying to avoid spending money, you'll have to get fodder heroes through other methods:
-using diamonds, scrolls, common scrolls and hearts to pull heroes at the tavern
-buy them using coins at the shop (especially at the barracks, where you spend coins you get for retiring the common heroes you can't really do anything with)
-trading soulstones (from tasks and other game modes, such as labyrinth)
So your question is less "how do I get to L+ asap" and more "how do i get lots and lots of diamonds and scrolls and soulstones asap" in which case:
-Do as many of the tasks and bounty hunts daily as you possibly can.
-Get as far in the campaign and king's tower for diamonds as you possibly can
-Do as much of the current events, arena of heroes, labyrinth, peaks of time, voyage of wonders, etc etc etc, basically all the main game and minigame stuff rewards you for playing
-Whenever you obtain a new hero, read their story right away. You get 100 diamonds just for opening the page (it's the little book icon on their status screen)
-Check the wall of legends, too. The more active your server, the more rewards you'll get really fast.
-Join a guild, do the guild hunt daily, use the guild coin rewards to buy equipment so you can make your units stronger and do everything else better
-Fill up your friends list as fast and as much as you can, to get as many daily hearts as you can
-Look up all the current bonus codes and use them. They'll usually contain diamonds, stones and scrolls. There's like five really good ones active right now
-Don't use diamonds to buy cards or soulstones at the shop. Pulling is much better odds-wise. Scrolls should be used ONLY for the faction you're currently working on (see next point)
-Check player guides to figure out how to make a good early-game team. This may be harder since you're focusing on only one hero right at the start, so i'd recommend picking a faction, not upgrading any of the legendary-tier heroes in it until you have eight of one of them, and using the other factions to construct your team as normal.
-If you can, borrow heroes from other players. You can apply to hire up to five mercenaries from people on your friends list. Player guides are good for figuring out who to hire (right at the start - Shemira is essentially the newbie carrier, good for the first fifteen or so campaign chapters)
-If you don't care about being extremely destructive, once you're in the home stretch, you can start consuming ascended-tier heroes to get your legendary-tier hero up that last little bit. This will make progressing in the campaign difficult and put the rest of your account WAY behind since it's a huge waste of time and resources, but if you only care about getting one hero to L+ then it's an acceptable trade.
The downside about doing anything in AFK Arena as fast as possible is that time is your resource - sometimes you literally can't really do anything without stopping playing and putting the game down for a while. It's easy enough to get an L+ in a short enough time, though. Good luck!