Chara -> Asriel
Asriel -> Chara

Sans -> Undyne
Papyrus -> Mettaton

Toriel -> Toriel
Asgore -> Napstablook

Undyne -> Alphys
Alphys -> Papyrus

Mettaton -> Sans
Napstablook -> Asgore

Toriel and Asgore were originally king and queen, after the death of their children (Asriel thought of the plan this time) and knowledge of what they really did, they both stepped down, not able to handle the grief.
Napstablook didn't really intend to take the throne, but they managed to help set order and everyone started looking up to them- they just go along with everyone else's plans, killing humans wasn't their idea.
Mettaton and Alphys moved to Snowdin for better reception and easier access to fuel for Mettaton. They work under Undyne and Sans together.

Both Undyne and Sans wanted to be in the royal guard- Undyne because she wanted to help monsterkind, Sans because he wanted to support his brother easier and had the skills to do it. The only reason they were really on "good" terms was because of Papyrus. After an accident (that I haven't thought of yet), Papyrus was badly injured and Undyne switched her focus into science instead to help Papyrus. Sans had also wanted to do the same, but Undyne assured him a better income would help Papyrus more than reserach.

Papyrus then got a TV show and such after being a cool roboskeleton dude
Undyne found the old tapes of the True Lab and made it her sworn mission to bring back the royal children. Asriel's dust was taken with Toriel and Asgore, but they couldn't get the body of Chara. Undyne found where they were burried (under a bed of flowers) and proceeded to start injecting it with determination. It seemed the experiments were unsuccessful, but the only one of the flowers that had started growing out of Chara (ew) came to life.

Chara simply goes as Red instead of Flowey the Flower.