There's a few corrections here:
-Ner'Zhul (the Orc) never worked for the Legion willingly. After he blew up Draenor, Kil'Jaeden snatched his soul and basically said "ok you help us conquer Azeroth or I torture you for eternity." NZ agreed, but KJ still didn't trust him so bound his soul into ice-encased armor, thus creating the lich king
-Arthas' arc was mostly manipluated by NZ himself to work towards his freedom from the Legion, not part of the larger Legion plan. Or at least, the Legion just assumed he would just be a powerful Scourge agent and help take out the Human kingdoms, but not much beyond that. Arthas going to Icecrown to put on the helmet (he already had Frostmourne since before he was a DK) was 100% him and NZ scheming, the Legion was actually actively trying to prevent that.
Woo an excuse to try to summarize Warcraft lore as succinctly and simplified as possible, I'll take it.
In ancient history Azeroth had one continent with the extremely powerful Well of Eternity at its center. Sargeras wanted to consume its powers and worked with Queen Azshara to enable the Legion's invasion. Malfurion and friends try to stop this from happening but are betrayed by Illidan. Everyone gets into a big fight, Illidan betrays Azshara and the Legion, the well explodes and splits the world apart, but the Legion invasion is stopped. Unfortunately Illidan has re-betrayed Malfurion and friends by recreating the Well of Eternity and exposing Azeroth to another Legion invasion. They throw him in jail to rot and plant the World Tree on the new well to help protect it from the Legion.
Now we fast forward to the Warcraft 1/2 timeline where Gul'dan/Medivh orchestrate (unintentional pun) the Horde's invasion of Azeroth on behalf of the Legion. The Orcs are eventually defeated by the Alliance, but in the process of trying to shut down the portals on Draenor they accidentally cause Draenor to explode and turn into Outland. While everything is exploding, Ner'zhul escapes through one of the portals but runs into Kil'jeaden who turns him into a suit of armor and deposits him in Northrend to work on the Legion's behalf as the Lich King.
Now we've hit Warcraft 3 timeline where the Lich King is using the Undead Scourge to facilitate yet another Burning Legion invasion. The Lich King manipulates Arthas into becoming his champion and Arthas goes on to summon Archimonde into the world. Archimonde and the Legion travel to Kalimdor to try to take over the Well of Eternity, leaving the Lich King and the Scourge in charge of the Eastern Kingdoms.
Illidan is freed from his prison by Tyrande so he can help fight the Burning Legion, but is later told by Arthas the location of the Skull of Gul'dan which will allow him to use the Legion's own powers against it. Illidan absorbs the skull and becomes the half demon we know today. Malfurion is horrified by this and banishes Illidan as a result. Archimonde is eventually defeated when Malfurion blows up the World Tree.
Now we're at The Frozen Throne expansion. As Illidan wanders Azeroth seeking ever more power, he convinces Lady Vashj and Kael Sunstrider to join him. Together they travel to Outland and usurp control of the planet from Magtheridon. Their success is short lived as they are immediately confronted by Kil'jeaden who has not forgotten about Illidan's betrayal of the Legion tens of thousands of years ago. Rather than kill Illidan, he conscripts him into the Legion once more and demands that he destroy the Lich King who has grown so powerful that Kil'jeaden can no longer control him.
Illidan returns to Azeroth and tries to use a spell to destroy Northrend and the Frozen Throne but is stopped by Malfurion who believes the spell will destroy Azeroth. The Frozen Throne is cracked by the spell however, which causes the Lich King's powers to begin draining and allows Sylvanas and the Forsaken to break free from the Scourge. Arthas travels to the Frozen Throne where he defeats Illidan in combat and merges with the Lich King.
And then we hit the WOW timeline. Obviously there's more detail to actions and nuance in terms of motivations but that's all the important bits I remember that would be relevant to this expansion in some way.
