Mihawk wears very decorative, ornate, black-and-red clothing. He wears a wide-brimmed hat decorated with a large plume, a long, black cloak, a cross pendant (which unsheathes a small dagger). a and carries a huge sword.
Mihawk is possibly one of the most serious characters in One Piece. Though he is the "swordsman above all the world's swordsmen", Mihawk remains grounded. Though he has obviously become jaded regarding the abilities of other swordsmen, he's not unable to recognize talent. In fact, his desire is to see Roronoa Zoro surpass him after having witnessed Zoro's incredible determination and will after their battle. He displays a sense of honor which other Shichibukai who have been introduced thus far seem to lack.
In the past Mihawk was also a rival to one of the most powerful pirates in One Piece, Red-Haired Shanks, which also speaks volumes about his power, but ceased their rivalry after losing interest in fighting a one-armed man like him.  He still has a neutral relationship with the pirate and seems to know how to track him down if he has business with the pirate. He and Shanks do manage to get along, regardless of the fact that the two are opposites in personalities. Shanks is also able to get Mihawk to join in when he threw a party to celebrate Luffy's coming, even when moments beforehand Mihawk had scorned how laid-back Shanks was.
Having lost one rival in Shanks, he now shares a rivalry with Zoro and is prepared to wait to any lengths and fight Zoro at his strongest form. Mihawk seems to be watching out for Zoro's progression and is delighted when Zoro gains his first bounty as well as amused by the Straw Hats' victory over Sir Crocodile.
Abilities and Powers
|“|| I feel sorry for you weakling. Even the greatest swordsman won't fight me. Because they can estimate their power and compare it to mine. Your strong will that wants to challenge me, does it comes from your brave heart... Or your foolishness?|
-Mihawk's response when Zoro challenges him
Mihawk is the strongest swordsman in the world; a monster superior to monsters, as one character calls him. Because of his status and power, defeating him is Roronoa Zoro's ultimate goal.
Mihawk is so strong, in fact, he is accused by some characters, as well as some fans, of having the power of a Devil Fruit; however, in the databooks, his abilities appear to be simply pure skill and power.
Mihawk travels around in a coffin-shaped raft with two green-flamed candles, a single black sail, and a single seat. The fact that he can traverse across the unforgiving oceans, as well as the Grand Line itself, in such a meager craft is yet another testament to his abilities. His first demonstration of power was to defeat Don Krieg's fleet of 50 ships and 5,000 men by himself just before a storm strike.
“ It's still too early for you to die. My name is Dracule Mihawk! Discover your true self, the true world, become stronger! However long it may take, I shall await you at the top. Surpass this sword! Surpass me! Roronoa Zoro!!!
-Mihawk's challenge to Zoro
He wields the Black Sword Kokutou Yoru, the strongest Sword in the world. It's an ornately decorated, man-sized (6 feet) sword shaped like a cross with a shiny black blade, which is curved at the end. It also seems to generate otherworldly purple streaks as it slices, as seen when he cut down Zoro and when he split Don Krieg's ship in half. Yet as well as having the ability to use it to slice up entire fleets of ships, he is graceful with the blade enough to knock bullets and change their course with little effort. 
The cross pendant worn on his neck is actually a small dagger. Though it appears to have no value, in the hands of Mihawk, it was able to defeat Zoro.