I have loremaster of all expansions.
Ill give you a few tips if you want to go down this long, long long road. I by no means came up with any of these but found the advice i was given and found to be of good value.
You are luckyy doing this now as it is a lot easier to get that achieve now. Back when i started playign the game we had to wait for RNG dropped quest starters before you could move on from that zone. No more.
Best class to do this in? Yeah sure, you can do this with any class you want. If you are doing this on a fresh 80 than all previous content will be trivial at best, boring and tedious at worst.
If you want to bore yourself to death playing wow than this is one achieve to do while totally over powered and over geared. You will be so bored as the quests wil never end. I forgot the actual count of quests but i know it is well over 8000 now.
If you want to have some fun doing this and dont care how long it takes then stop xp on one level before you move onto next expansions.
Best class to do this in, in my opinion, is a feral nelf druid or a rogue. Reason why is stealth and the ability to bypass the multitude of mobs you will meet on the way to completing those quests. Being able to skip stuff and not fight when doing over 8000 quests will help speed things up.
If you go the road of doing this on a max character then you will need some things.
A foam sword. Its a toy. Some quest mobs need to be tagged and not killed and at 80 you will one shot anything and everything the second you hit them. Cant do that. It will void the quest and you will have to start over. So tag the mob with the foam sword first. Which in itself at 80 will put a nice dent in them.
Owning three hearthstones helps. Being a druid with the valshara teleport helps when all your HS are on CD as the teleport only has a 2 min CD. Being a druid helps with instant flight form as well.
Wowhead will be your firend. You wont believe the amount of times you willcome here to see how to do a quest.
Good luck. When completed you get thsi cool tabard that has a big yellow ! on the front. You can see it in my avatar.
One thing i will say is that you have no idea what you are getting yourself into, yet.