Spawn in the world as usual and play as usual on Hard until at least two nether portals can be created, once you have enough obsidian then the game becomes "hardcore".
Each life is assigned a color (Red, Orange, Yellow, Green, Blue, Purple, Black) more colors can be used, but I'm using the color wheel for my game. The first base area is Red, which is essentially everything that you do between spawning in the game and the first post portal death. The Red base is the nexus for the entire first round of the game, all other cities will span out from it.
The Red player will play and build their city like normal, but once they die, that is it for the Red city and it's surrounding region.
Between players the game is set to peaceful so that the next player can be set up. In order to do that there are a couple things to do.
1) Place a grave where the previous player died (optional, but fun)
2) It's also fine to take the possessions that dropped upon death and put them in the chests back at the base (again optional)
3) Once ready to start the next player, equip them with 2 or 3 picks, 2 shovels, 20 obsidian, flint and tinder, and a stack of torches. This equipment is only to get to the next starting location, dye in the color of the next character can be brought as well.
4) Enter the nether and start digging in any random direction for as far as you can on the picks. If your portal is in an open area then just run a distance that you feel is far from your original spawn.
5) Once you've travelled far enough, set up a portal and re-enter the overworld. In order to make things very difficult, place obsidian over the entrance to the portal in the outside world so that there is no going back. BUT make sure you can get to the surface before doing so!
6) Once on the surface and you know that you can play like normal, throw away any items that you brought to get you to the surface from the original spawn location. picks, shovels, flint, torches, etc... This should start you off with nothing, just like you would if starting a new world.
7) When day begins in the game, set the difficulty to Hard and play like normal.
One extra rule
All base structures must be built above ground, meaning no digging into the side of a cliff to make a house. Mob traps and other minor structures are fine to make undergound, but nothing that you'd consider a base.
In order to find the old locations you are required to make a diamond pickax so that you can open the portal and return to the original spawn, however you aren't allowed to use the portal just yet. In order to use the portal you need to find the original portal on the surface. Basically exploring your way back to the Red base.
Once you've reached the Red base then you are free to use any materials that are stored there and the portal to get between the two locations. But I would recommend not using the nether as it makes it kind of easy.
The Black character is essentially time. After going through all of the colors you will have Mr. Black to play with but his objective is different, instead of building a new city, his job is to destroy the old ones via TNT only. By making or using TNT from the previous cities he destroys their buildings with explosions, basically to mimic the effect that time and the elements would have on a ruined structure over time.
There are a couple ways of playing this character:
1) Go around destroying all the other cities, regardless of death until they have all been destroyed.
2) Destroy the other cities, and kill yourself in the explosion so that you don't see the effects of the ruins, making it a surprise for the future.
3) Play Hard like normal and work towards the destruction. Playing like you would with any of the other characters, except that you can travel to the old cities as soon as you want. But once you die then the next Red player takes over.
After you're done having fun with Mr. Black it's time to restart settling the world again, just like rebooting the matrix. I guess Mr. Black is Neo =)
Anyway... Enter the starting portal just like you would for any other character, except this time you want to go at least twice as far away from the original spawn this time so that there is plenty of room to re establish the world. Then play like normal.
If you find a ruin during play then you are able to take what is ever remaining there for your own use. It's suggested that you don't rebuild the structures because they are supposed to look destroyed, but that's up to the player.
Over time the world will be settled and ruined over and over again creating a huge RPG style world. The fact that you're starting over all the time is akin to a Hardcore mode, but unlike "the hardcore mode" where the world is deleted, your world keeps growing over time, but for you it's still a challenge to try and stay alive.