Why did arthas become king of the undead?

Its not a matter of Why he did its a matter of howhe did it wasn't his choice, it was the Lich King who chose him long before he was even born apparently to be his champion, so with Mal'Ganis's aid the Lich King was able to draw Arthas to Northrend and to Frostmourne which ended up turning Arthas into what you see today, and it wasn't actually until he fused with the Lich King in WC3:TFT that he became "king of the dead" Hope that answers ur question =D