DISCLAIMER: This guide isn't meant to compete with the speed you would level with through questing, but rather server as an alternative for those who don't like questing (or are just tired of it!).
There are 2 alternative ways that I personally always use to level my characters. I've never been a fan of questing, so in the past I've always ended up looking for spots with great respawn rates to farm at.
However as most of you already know, Blizzard has made it possible to gain experience through Battlegrounds and professions. It was meant to serve as something to help the leveling process on the side, but I've successfully used these to level almost 2 characters to 85 (one is still at 82).
BATTLEGROUND METHOD:
It's quite simple, and is great for those who afk-bot or those like me who like to do work close to their computers and just jump in-game every now and then to prevent the AFK debuff from happening. I'd go into detail here, but I found a free video from Skill-Capped that does a wonderful job explaining the experience mechanics of this IN-DEPTH. The video can be found HERE.
PROFESSION METHOD:
Now this is a method that most people overlook, but actually serves as an amazing alternative to questing.
This involves the professions HERBALISM and MINING.
Much like the mechanics from battleground, each time you herb or mine you'll be granted a small percentage of your total experience required for that level as experience. For example, at level 84, herbing a Twilight Jasmine granted me ~6000exp per node! That works out to be 1500 nodes for 9 million experience, and I was probably hitting ~5 nodes a minute. That would mean if I could keep that up, I'd level from 84 to 85 in 5 hours from just herbing! If I had mining in addition to herbalism. it would go even faster!
The amount of experience you receive per node depends on the level of the node you're farming (Orange > Yellow > Green). If possible, it's best to farm Orange nodes as much as you can, but if someone else is farming it, Yellow nodes always serve as a great alternative.
As a HERBALIST (I didn't have mining), my favourite spots were in Twilight Highlands, Uldum, and Deepholm, farming Twilight Jasmines, Whiptails, and Heartblossoms, respectively.
For the most part, if I'm the only one farming the zone, in a full circuit around the zone, the spawn rate is just about fast enough to let me continuously circle the zones to my desire.
However, if I'm competing with someone in that zone, I like to farm Deepholm for Hearthblossoms, as the respawn rate there is ridiculously high and can respawn fast enough for multiple people to be running circuits around the zone.
To find which circuits are best, I used Wowhead to find out where they were frequently spawned, and just ran circuits around those areas.
As a MINER, I believe Deepholm is the best option in combination with HERBALISM.
Or, if you're a Druid or have a Sea Horse, you can use the technique found here for similar results as well.
Deepholm also serves as a great alternative because if I'm farming any other zone and I find out that someone has joined in to compete with me and I want to continue maximizing my experience, I can use a Potion of Deepholm - Spell - World of Warcraft to port myself and continue herbing.
BONUS (GOLD!):
As a bonus, all gathered materials sell for extremely high amounts at the moment. For example, when I was doing this at the beginning of the expansion, Whiptail stacks were selling at 1000g a stack on my server, a stack which I could farm in under 5 minutes using this method! However, the prices aren't as high now, but a stack of Whiptail for example still sells for an east 150-200g on my server. If you use this method and your SOLE leveling method, I promise you that you won't be missing out from the gold from quest rewards :P (but you might miss the reputation!).
All in all, I feel that the PROFESSION METHOD serves as a good alternative way to level if you don't want to quest, but shouldn't replace questing altogether (because reputation is important still!). The BATTLEGROUND METHOD could also still be used, perhaps weaved into the PROFESSION METHOD, as after a game you can come back out and all the herbs/ores would be refreshed for you.
I hope this guide was useful for those who just despise questing, people like me.
Cheers!