I am the eldest of the first,
Said to have never died.[/quote]
Dwarves were made before the elves by Aulë, but awoke later
And he's called the deathless
QUOTE
I founded a great city,
Seeing stars as a sign.[/quote]
He founded Khazad-dûm after seeing stars reflected in the surface of a lake which he took as a sign
QUOTE
My line ruled throughout history,
Though not forever in that realm.[/quote]
Durin is the Founder of Durin's Folk and all dwarved hail from him, so they would rule throughout history, but at some point Moria was lost, and they obviously didn't rule there anymore then.