We learn through a report by Duke Senior's men that he is given to a chronic melancholy as they humorously relate how he was very depressed and sad over the death of a stag that they have not willingly shot.
First, the Jaques you are referring to is attending on Duke Senior, not the Jaques who is Oliver's other younger brother.
There are a few reasons why Jaques could be sad. One, he is a high class man of the court, but now he is living in the woods like a savage. For an older man of means who is used to a comfortable life, this sits hard.
He likes it. He loves an argument. Look at his conversations on love with Orlando. They speak volumes about how his reasoning maintains that there is no real love in humans. Also, for more enlightenment on his character look at his description of man in all his ages. Hardly very complementary. Finally, look how he enjoys making fools out of others, as in calling the men into a circle of fools to hear him.
But why in the midst of all of this frivolity and love-festing, would Shakespeare have a character like him? Imagine not having him. Secretly, we all agree with Jaques. Man does make a mess of life.