The story of Arthas takes place during the Reign of Chaos... Arthas, who was a Paladin, quells the orc uprising and then heads north to stop the scourge. Medivh stops him and tells him he needs to go to Kalimdor to defeat the scourge, but Arthas doesn't listen and instead goes to Stratholme find the rest of the infested grain that was causing people to turn into undead. There, he meets Malganis, the Dreadlord, who says that he will meet Arthas for a final battle in Northrend. Arthas then heads north and encounters a Dwarven search party that was looking for the sword Frostmourne. When he finally finds Frostmourne, he finds out its cursed but still takes it anyway.

Eventually, Arthas finds Malganis, and Malganis tells him that the voice coming from Frostmourne is from the Lich King. Arthas then kills Malganis.

After staying in northrend for a few more months, Arthas returns to Lorderon and kills King Terenas. Then he runs away and begins to build an undead army.

It isn't until after he kills Terenas does Arthas begin practicing necromancy and being a death knight. Keep in mind though, he's not undead. He's alive, but Frostmourne has captured his soul.

To make a long story short, Arthas eventually heads back to northrend, and defeats Illidan and returns to the Frozen Throne. He uses Frostmourne to release Nerzhul, and puts on Nerzhul's crown, fusing his soul with Nerzhuls, and becoming the Lich King himself.

At this point, adventurers head to northrend to defeat Arthas, the new Lich King. Trion Fording destroys Frostmourne and releases Arthas's soul. Frostmourne also releases the souls of all the other people killed by it, including Terenas. Arthas then dies.

So Arthas was first a Paladin, then a Death Knight, and then the Lich King. Only when he was the Lich King did he have the powers of a Lich.