Give your hero noble origins. Usually, epic heroes descend from gods, great kings, or ancestors blessed with outstanding qualities. In modern fiction, some epic heroes have an ordinary pedigree, but making your hero part of a long line of greatness definitely feeds in to the sense of heroic destiny.
Have your hero face a great challenge. Cuchulainn, for example, faced a whole invading army singlehandedly. Hercules had to accomplish twelve tasks thought to be impossible. Other heroes had to face difficult voyages, unite warring kingdoms, or journey through death.
Have your hero's deeds be prophesied. If people believed that your hero would come to accomplish whatever great task he is charged with long before he was born, it adds to the importance of what he is doing. It is as if all of history has lead but to the hero's journey.
Give your hero awesome powers. Many epic heroes are blessed with superhuman strength or near invulnerability. Others can speak the language of birds or look into the future. Some are just extremely clever, charming, or lucky. You really can't overdo it with an epic hero, provided that you can tie his abilities into the story.
Give your epic hero a substantial character flaw. Epic heroes need to change, and in order to do that, your hero needs some faults. Often, it is the hero's flaws that start his journey in motion. For example, Ulysses had hubris. If he hadn't mouthed off to the gods who he thought he didn't need, he wouldn't have offended them. It would have been smooth sailing back to Ithaca, and there would have been no Odyssey.
Have your epic hero meet friends and allies along the way. These allies can be mortals or supernatural in nature, but they should lie outside his previous realm of experience. An epic hero usually learns from seasoned teachers, tricksters, love interests, or powerful beings of some sort.
Have your epic hero return home, changed. An epic is always cyclical in nature. The hero accomplishes what he wants to do and either dies, having lived up to his tragic fate, or returns, older and wiser.