Mihawk is possibly one of the most serious characters in One Piece, very little if nothing at all seems to suprise him. 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 able to recognize talent. In fact, he has a desire 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 seem to lack, with the exception of Bartholomew Kuma and Jinbei.
Mihawk is selective when it comes to fulfilling his "duties" as a Shichibukai as evidenced by his history of poor attendance at Shichibukai summons, as well as his decision to hunt and harass some pirates over others.  Zeff speculated that the reason Mihawk attacked Don Krieg and his men might have simply been because they disturbed his nap. Mihawk admitted that the only reason he followed them all the way back to East Blue from the Grand Line was to kill time.
EnemiesIn 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 Shanks and seems to know how to track him down if he has business with his old rival. 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.
Mihawk also made himself to be a dangerous enemy of Don Krieg in their attempt to conquer the Grand Line. However, this issue seems to have resolved itself with Krieg's defeat to Luffy. Nevertheless upon sighting Mihawk, Krieg's anger led him to attempt to take him out despite witnessing the event between him and Zoro mere moments beforehand.
Abilities and PowersMihawk 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 defeating Don Krieg's fleet of 50 ships and 5,000 men by himself just before a storm struck.
WeaponsHe 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 with the slightest bit of pressure, and was able to stab close to his heart. 
He's able to attack enemies with powerful slashes made of a huge stream of energy or compressed air (it's unknown yet) like they were giant projectiles, like Zoro's Pound Hou techniques or the different variations of Rokushiki's Rankyaku.
HistoryMuch of Mihawk's past remains a mystery. However, he once saw a strong rival in Shanks. Their battles were said to shake the whole of the Grand Line and Mihawk often sought Shanks out for duels. After Shanks lost his arm ten years ago, Mihawk ceased seeking him out for duels. At some point Mihawk earned the title of "World's Greatest Swordsman" as well as joined the ranks of the Shichibukai, however when both these happened is unknown.
Confrontation at the BaratieMihawk had been responsible for single-handedly destroying Don Krieg's entire fleet, save Krieg's flagship, during his failed attempt at entering and conquering the Grand Line. The destruction of Krieg's fleet drove the pirate from the Grand Line. He then followed Don Krieg to East Blue, to the Baratie and destroyed their flagship as well.
Mihawk strikes Zoro
While at the Baratie, he encountered the Straw Hat Pirates, and Zoro immediately challenged him. Mihawk was able to block all three of Zoro's swords with the tip of his necklace blade, finally stabbing him in the chest with it. Mihawk was so surprised by Zoro's spirit, however, that he granted him a face-off against the Black Sword. Mihawk won flat-out, shattering two of Zoro's three swords (all but the Wadō Ichimonji). 
Zoro, admitting defeat, allowed himself to be cut down by Mihawk face-to-face, like a true swordsman. Mihawk accepted this offer and cleaved open a massive diagonal cut across Zoro's entire chest , but held back just enough to allow Zoro to live. Luffy had already struck out at the Shichibukai unsuccessfully when Zoro called out to him and proclaimed that no matter what he would never lose again. Afterwards, Mihawk told Zoro his full name, and gave him a challenge. 
“ It's still too early for you to die. My name is Juracule 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
After hearing Zoro's goals and giving him a challenge, Mihawk asked Luffy what his goal was. When Luffy told Mihawk it was to be Pirate King, Mihawk told him that was a more foolish dream than challenging him.
With the events against Zoro over, Mihawk decided it's time he headed back to the Grand Line for a nap. Don Krieg took offense to the Shichibukai attitude towards his men and launched an attack at him. Mihawk simply called out "You fool!" and in one final strike struck at Krieg's ship again, causing the already wrecked ship to fall further into pieces. When the dust cleared Mihawk was no where to be seen.
Shanks and MariejoisSoon afterwards, Mihawk is seen visiting Shanks to bring news of Luffy's coming. Then later on he takes part in a meeting with the Marines and two of his fellow Shichibukai, Donquixote Doflamingo and Bartholomew Kuma.
Current EventsApparently, he has answered to the call of Shichibukai to battle Whitebeard and is joining the war, later shown dining with the other Shichibukai, except for Jimbei and Boa Hancock.
He is next seen at Marineford, alongside Bartholomew Kuma, Donquixote Doflamingo, Gecko Moria and Boa Hancock, ready for the battle ahead.
When Aokiji froze the two tsunamis that Whitebeard caused, Mihawk was one of the first to attack, wishing to measure his strength against Whitebeard or one of his crew. His attack of the "world's strongest slash" though was stopped by Jozu; Mihawk's attack had not damaged him at all and it is unknown as whether or not Mihawk has an attack which could cut through diamond yet.
He shows slight amusement upon Luffy's arrival at Marineford, remarking that the Straw Hat never failed to make things interesting. When Sengoku reveals the fact that Dragon is Luffy's father, he only comments that such a revelation isn't really surprising.