The hero is
Oh woe is he! How cruel is fate!
He pouts and mopes in this hour late.
After his failed attempt to climb Mount Olympus, he spent the rest of his life a blinded hermit, wandering on the Plain of Wandering (original, I know). "This hour late" refers to the twilight of his life. His fate is cruel because he was a massive hero reduced to nothing.
The subject of some cruel god's whim;
Death the gods have in store for him.
Bellerophon was sent by King Proteus of Tyrins to King Iobates of Lycia to die. Iobates feared the wrath of the gods for breaking the Greek laws of hospitality which forbade him from killing a guest, so he sent Bellerophon on an impossible quest: defeat the Chimera. After defeating the Chimera, Bellerophon did not receive credit from the king. Afterwards, he was sent on even more quests, most of which were intended to kill him.
Suddenly a beast of legend!
Our hero's sentence swift deadened.
This is the Chimera, a beast of legend that breathes fire. He managed to defeat the Chimera by putting a block of lead in its throat; the Chimera's fire-breath melted the lead and then it solidified, blocking the passage of air.
As it roars, he recovers. Run!
Before his hide, as well, is done.
When he first tried to attack the Chimera, he couldn't get close enough to damage it, even riding on top of Pegasus. The Chimera's fire-breath will literally cook Bellerophon's hide (and make it well-done!)
Fleeing there, to the city pale;
News for the king, no time to fail.
With men and arms they slay the beast.
Huzzah! Huzzah! Now time to feast.
After defeating the Chimera, Bellerophon went to King Iobates with the news.
What is this feeling now unleashed?
Some kind of bond 'twixt him and beast.
He feels altered, not quite the same.
The men surround him, call him names.
Bellerophon tamed Pegasus, and the two created a bond, as rider and noble steed.
All at once, the skies asunder.
A booming voice cracks like thunder.
Illusion? No: a beckoning;
to mountain high 'ere reckoning.
Bellerophon believed that his fame gave him cause to ascend Mount Olympus. He believed that Zeus called to him, beckoning to ascend to the top of the mountain, where only the gods and their servants lived. Zeus sent a horsefly to bite Pegasus, causing Bellerophon to fall down and be injured for life.
From here does his adventure start;
Wisdom, cunning,and strength in part.
Though truly only one of those
This hero of our story chose.
Bellerophon showed wisdom and cunning in his ability to tame Pegasus, defeat the Chimera, and vanquish the Solymi and Amazons (he dropped boulders on them from Pegasus). He showed strength when he prevented an ambush by the pirate Cheirmarrhus and defeating palace guards sent against him. However, he abandoned wisdom and cunning when he attempted to ascend Mount Olympus.