Many sea names were mentioned before, and @jafe even pointed to
the Wiki page of the Pirates of the Caribbean: At World's End, where the 9 seas are listed.
What is different in my solution is that I think
We need to match these exact seas to the riddle, and find the one that is left - the traitor.
The first of all is the chief of all
Hidden by the songs of death
I think this most obviously refers to the Black sea, as black is the color of death.
The second lord brought with him a dove
And the largest army of the bunch
The dove is the symbol of peace, wich aligns well with the Pacific Ocean. Also, it is the largest sea. (Also found by @jafe)
The third of such is the most known of such
Named by a foreigner to the original one
The tenth pirate hint that this is the sea which was found by Columbus - the Caribbean Sea. It is also the most well-known, because of the movies. (Also found by @jafe)
The fourth lord is cruel and lost
And dragged the Titan down to the abyss
The titan Atlas shares the name with Atlantis, the lost city, which dissapeared in the Atlantic Ocean (what a coincidence - also the same name). (Also found by @jafe)
The fifth pirate is known as Helen, aged yet classy
Some may call her Tethys, others, a hybrid
The Tethys Ocean is the ancestor of the Mediterranean Sea, which i think is the most classiest of the seas. Also, the presence of Greace could explain the Helen reference.
The sixth one birthed the iron with a feet of clay,
Known by the Greeks as horse of superiority
I have to admit, this was the one I had as a leftover from the nine seas - the Caspian Sea. The ironworking originated from this are, and the Trojan horse's breed may be Caspian horse.
The seventh pirate's a distant relative to Heracles
Though in possession by the Roman army
The Adriatic Sea, was ruled by the Romans, and it is very close to Greece and the Mediterranean Sea, which can explain the Heracles reference. (Also found by @jafe)
The eighth pirate reigns up not down
Don't let it's waters deceive it's name
This will mean the South China Sea, which has South in his name, but is actually on the Northern Hemisphere.
The ninth lord reigns down not up
The waters describe it's name
I think he is the traitor - the Dead Sea, which is not on the list of the nine seas. He reigns down, in the empire of death, and clearly goes very literally with the last question: "Which pirate is to be dead?"
The tenth pirate may have met with Moses
Columbus looked for him but found the third instead
Moses parted the Red Sea, which is part of the Indian Ocean. It is also clear that Columbus originally looked for a way to India.
Summing that up:
The traitor is the ninth lord, the lord of the Dead Sea - he shall be dead.