Frodo can be interpreted both as an unwilling hero, but also as a willing hero, determined in his goals and actions.
For the unwilling hero, see the early parts of the novel, where Frodo is essentially coerced into becoming the Ringbearer by virtue of inheritance. He is not expecting this sort of mission, nor, once informed of his burden, in any way eager to accept it:
"...I am not made for perilous quests. I wish I had never seen the Ring! Why did it come to me? Why was I chosen?"
To tell the truth, he was very reluctant to start, now that it had come to the point.
Frodo, as seen here, was compelled to take up the task more out of necessity than out of choice. Despite his yearning for adventure, spurred by Bilbo's own recounts, Frodo is a classic homebody, finding his "boring" life more desirable after he is forced to abandon it.
However, throughout the book Frodo also shows signs of being a willing hero; he is
An overwhelming longing to rest and remain at peace by Bilbo's side in Rivendell filled all his heart. At last with an effort he spoke, and wondered to hear his own words, as if some other will was using his small voice.
"I will take the Ring," he said, "though I do not know the way."
(Tolkien, The Fellowship of the Ring, Google Books)
It is not that he desires to take the Ring to Mordor, but that he feels that it is his duty, since it was passed specifically to him. Despite this reluctance, Frodo continues on his quest when others would have failed -- and where many do fail. He even makes the decision to abandon the Fellowship itself, as he sees how the Ring affects their minds and causes dissent; he believes that the quest is more likely to succeed if there are no others to be tempted by it.
Frodo is a complex character, and should be studied in depth before any single or superficial character description is applied.