The best way to be able to see a glimpse into Hamlet's romantic love is if we look at his relationship with Opehlia. Opehlia and Hamlet had been romantic; when she confronts him at the bidding of her father, Claudius and Gertrude, she tries to give him "remembrances of his," meaning, little love trinkets and gifts that he had given her. As she tries to give them, she explains that when he gave them to her, he also gave
"words of so sweet breath composed as made the things more rich" (III.i.97).
So, he gave her presents, and spoke to her of beautiful and sweet things. It seems that Hamlet was quite the romantic: giving presents, perhaps reciting poetry and sweet nothings to the fair Ophelia. Earlier, when she was talking to her father, she admitted of Hamlet that "he hath...of late made many tenders of his affection to me" and "importuned me with love in honorable fashion" (I.iii.98-111). So, he had been giving her quite a bit of attention, honorably, with "all the vows of heaven" (I.iii.114). He seems to have been a very intense, gentlemanly, devoted love interest in Ophelia's life, courting her in a very proper manner, but giving her every indication of his affection.
Later, at Ophelia's grave, he is distraught, jumps into her grave and proclaims,
"I loved Ophelia: forty thousand brothers could not, with all their quantity of love, make up my sum" (V.i.292-4).
This sincere and passionate declaration indicates that Hamlet felt deep and romantic love in his life, of such an intense depth that he compares it to the strength of 40 thousand men.
All in all, Hamlet seems to be quite the romantic guy; passionate, deeply capable of feeling in love, and showing those feelings well through words and gifts.