Hamlet Act I scenes i-ii

The first two scenes of Hamlet do not give much background information of the main characters and the relationship between characters, but rather leaves the reader to figure out the backgrounds and relationships of the characters. The reader learns in scene two that Hamlet is the prince, and that King Claudius is both his uncle and now father. In that encounter, Shakespeare exposes a sliver of the type of relationship the two have. Hamlet's relationship with his father, which we cannot judge through dialogue, seems to be apparent through his heavy mourning over his father's death. Hamlet's encounter with his mother, Queen Gertrude in scenes i and ii is breif, therefore much speculation is required.
In the middle of the second scene (line 64), Hamlet is introduced as King Claudius' son and cousin, but is actually really the king's nephew. By Hamlet's response in line 65, it can be concluded that Hamlet is not very fond of his uncle. Nevertheless, King Claudius tries to comfort the mourning Hamlet and tells him that everyone loses their father at one point in time, but also gives him some sharp words that questions his character; specifically line 94 as King Claudius calls Hamlet's mourning "unmanly grief."
In the same scene, Hamlet speaks with his mother as she asks him to stay with her and King Claudius and not to go back to Wittenberg to continue studying. He does not place any words of affection or endearment towards her, and responds with "I shall in all my best obey you, madam" (line 120). This may be a cultural custom, but I see this as potentially a somewhat weak relationship between Hamlet and Queen Gertrude. In the following sequence, Hamlet's soliloquy, he appears to be upset with his mother for, in his eyes, betraying his father's love for her by remarrying King Claudius (lines 145-156).
At this point of the point in the play, it is premature to form any definitive opinion on Hamelt's relationships with King Claudius and Queen Gertrude, but the foundation has been set. It appears that there is hostility between Hamlet and King Claudius, and maybe even contempt. As for Hamlet and Queen Gertrude, Hamlet appears bit disappointed in his mother and perhaps upset in her perceived quick recovery from her former husband's death.