You keep one character as a "key", for every raid boss that you want to farm, you get your key character saved to one boss before it. So if you want to farm Ji-kun, Elegon, Ragnaros, and Lich King, you get your key character raid locks with Magaera, Spirit Kings, Fandral, and all bosses before LK killed (on 25N). Each week, you will get on your key character and extend the lockout. Even if you don't run, you should extend your lockouts as often as possible, so you don't lose them.
Then each week, you will put your farming character(s) in a group with your key character. If your farming character hasn't been saved to any bosses, then the game will auto-clear the raid to match the instance lock of the furthest progressed character in the group. Including offline characters no matter where they're parked, and also the 'group' will have a raidid assigned to it, but that can be reset by doing "reset all instances" by whichever character is leader, when nobody (online or offline) is inside the instance. If your farming character has killed early bosses in that instance, then the key character will have to be raid leader (offline is ok) when your farmer enters the raid. It takes a few minutes to auto-switch leaders, so this is time consuming and tricky.
What I do, when I farm, is get all my characters (farmers and the key) into one raid group, and then take each farmer on a raid-tour. When a farming character is done with all its raids, I toggle raid size/unheroic (sometimes its required, sometimes not, always a good idea), and then reset all instances with that character (since its still the lead). Then I drop group, and relog quickly to the next farmer. When I log onto the new farmer, I toggle raid size/unheroic, because the interface can be inaccurate if you don't.
There are several ways to get all your characters in the same raid group.
1) Have two accounts. Your second account doesn't even have to have any leveled characters on it, a level 1 will work. Although a character does have to be level 10 to easily convert to a raid group, there's a /slashcommand to let a level 1 do it (I haven't tried it). Alternatively, you can just pass lead to a highbie in order to convert to raid.
/run SetAllowLowLevelRaid(true)
2) Have a friend invite all your toons for you. If you have a friend who doesn't mind, this is the fastest/easiest. Letting them use your key-character for their own farming after you're done is a good way to repay their help. For their convenience, you can combine with method #4. They just hold lead until you're ready.
3) Use the custom groups in LFG tool. After you make a group, you can switch characters and request invites, then switch back to the lead and send out invites, then switch back to accept invites. You can do about 2-3 characters (request on all first, then invite on all, then accept on all). The leader will automatically change to whomever is online when the original leader has been offline for a short while. Mine usually happened on the third character. How many characters you can do per wave depends on how fast you log in/off. You have 2 minutes after request to get invited, and I think 2 minutes after each invite to get accepted.
4) Use the custom groups in LFG tool, using auto-invite. This is a lot faster, since you never have to go back to your original character. But this can be hazardous, as you can get hoppers. And because of the timing of the "auto-switch" for the raid leader, a hopper can end up with lead. Put in your description that you don't want hoppers, without using the word "hop". Courteous hoppers will avoid it, but some are jerks. And some have addons that are jerks for them. Even though hopping has been severely nerfed, they keep trying. And put the group in raids, not custom. And choose a boring raid.
Different expansions' raids sometimes work a little differently, especially when it comes to heroic raids.
- Vanilla and BC, this method does not work at all. But since you don't need to clear tempest keep for the mount boss, this isn't much of a problem.
- For WoLK, this method works, but you will need to set to heroic (while inside works, if that character has done 25N) for LK, and set raid size to 25 (when outside) for both LK and Yogg.
- For cataclysm, this works for Firelands. If you are doing both mount bosses, you only kill giant, spider, hunter, balrog on your key character.
For dragon soul, you cannot do heroic Madness using this method (the other mounts work). To avoid accidentally ruining your key character's raidlock, clear it using normal mode. It is possible to do two key characters, one for each boss, but it's tricky and time consuming, so it's probably easier to just only farm one mount at a time (or spend the time killing all the middle bosses each time).
- For pandaria, this works for MSV and ToT. Again, it's possible, but tricky and very time consuming to use two key characters for the two mount bosses in ToT. For SoO, this will not work, you need a full clear on mythic for each garry kill.
- warlods. I haven't tried yet, but if I remember correctly, all the mounts are on mythic, so you'll need to do full clears each time.
If you are also farming MoP bosses, you can park your farmers at the exits to ToT or MSV when you drop the raid group, and kill them after you're done with the raid instances.
I hope I haven't forgotten everything. And I hope I was able to explain well enough (I may end up editing for clarity). Good luck, and don't be discouraged. Random can be clumpy, so you may end up with long dry spells. I had a real long one, and then got two mounts within 10 minutes of each other.