I did this on SoM and it was really fucking fun and also almost gave me a heart attack a few times. Here's the TL;DR for anyone considering this.
Get the addon, you have to have it before you create your character for it to count.
Addon | Classic Hardcore
Rules and tips:
-you cannot group*
-you cannot trade (not even conjured items)
-no mail or AH
-tradeskills allowed, you can wear whatever you craft but you have to farm the mats yourself
-your only cash income is from what you loot or from quests, so skinning is a solid trade to get. pair with tailoring early on to make yourself some starter bags
-questing is allowed and encouraged for both the exp and gear rewards. you may want to grind mobs at some points for exp. generally, you want to overlevel the quests.
-if there's any doubt in your mind that a quest may be too hard, skip it and move on (or wait a few levels). skip the elite quests.
-join a guild if we have one, or else just some random guild. The addon broadcasts whenever someone dies which is hilarious and leads to a wall of F's. "Timmy the Paladin has died to Hogger at Level 12". "F"
* - exception for grouping, you're allowed to group for one run of each dungeon (have to be in the level range of the dungeon).
Real talk, do not attempt any dungeons with randoms. So many people died in my SoM guild at level 36 or whatever due to a bad group. Stick with pals or skip the dungeons.