Romeo thinks a great deal about fate, even from the beginning of the play. In fact, we see him warn his friends that crashing the Capulet's ball is a poor idea, even declaring he had a dream predicting his own death as a result of attending the ball. We see this when Romeo declares:
And we mean well, in going to this mask;
But 'tis no wit to go.
I dreamt a dream to-night.
Some consequence, yet hanging in the stars,
Shall bitterly begin his fearful date
With this night's revels and expire the term
Of a despised life, clos'd in my breast,
By some vile forfeit of untimely death. (I.iv.50-53, 114-118)
In other words, Romeo is saying that he dreamed that some ill-fate, or unfortunate event, would happen as a result of crashing the ball. Namely, since Romeo speaks of the life in his breast expiring, Romeo is declaring that his dream predicted his own fated and young death. Hence, we see that Romeo has always felt governed by fate.
However, later in the play, we do, indeed, see Romeo's struggle with fate change his personality. During his exile, Romeo remains hopeful and has a second dream about Juliet waking him from a state of death, which gave him hope for the future (V.i.1-10). However, immediately after this dream, Romeo learns that Juliet has been found dead. It is at this moment that Romeo feels absolutely defeated by fate, and his personality changes from upbeat and hopeful, to haggard. His personality changes so much that when he encounters Paris at Juliet's tomb, he begs Paris to let him die in peace and begs Paris not to challenge him to another fight, saying:
Good gentle youth, tempt not a desp'rate man.
Fly hence and leave me. Think upon these gone;
Let them affright thee. I beseech thee, youth,
Put not another sin upon my head
By urging me to fury. (V.iii.59-63)
In other words, while Romeo was once a peaceful person, he feels so desperate from what fate has bestowed upon him and so desperate to put an end to his own life by Juliet's side that he will kill any one who gets in his way. Hence we see that fate has changed Romeo from a peaceful, hopeful person to a haggard, violent person.