In Hamlet Act I, what are quotes that show family tension, what the tension reveals, and how it impacts character develop and another family nexus?
There is more than one tension in Hamlet's newly restructured family comprising his paternal uncle Claudius as his father and King with his mother Gertrude as his aunt and second-time Queen by virtue of her marriage to his uncle. This description of his family epitomizes one of those tensions: his father, the King's brother, has taken the King's throne, married his wife the Queen, and added "Father" to his existing designation of "Uncle" in relation his nephew and son Hamlet. Hamlet would like to be mourning the death of his father and King but instead he is mourning both the death of his father and the marriage of his mother while also raging against the usurpation of his father the King's throne by his Uncle Claudius who is now his father and his King and his mother's husband. There is enough there for plenty of tension.
Another source of tension, which is the tension that fuels the drive in the play, is Hamlet's continued mourning while Claudius is needfully desirous that Hamlet stop mourning his father's death and start celebrating his mother's new husband, the country's new King and his own new father. This is what Claudius means while referring to clouds that "still hang on you":
CLAUDIUS: But now, my cousin Hamlet, and my son,--
HAMLET: [Aside] A little more than kin, and less than kind.
KING CLAUDIUS: How is it that the clouds still hang on you?
HAMLET: Not so, my lord; I am too much i' the sun.
QUEEN GERTRUDE: Good Hamlet, cast thy nighted colour off,
Claudius needs Hamlet's acceptance because Hamlet is much loved throughout Denmark and beyond and, without his contentment and acceptance of the new order, there is a chance of political upheaval erupting.
It is this tension over Hamlet's continued mourning and grief that is the major impetus to the progress of the play. It reveals, for one thing, Claudius's manipulative and corrupt nature that leads him even to send Hamlet to the hands of those who would kill him. It also reveals the inner conflict Hamlet undergoes in trying to choose between two moral goods: to stay away from ghosts because of his Christian Reformationist leanings (Wittenberg University was the center of Reformationist teachings) or to avenge his father's death. Ophelia's conflict of being torn between her love for Hamlet and her duty to Claudius is also revealed.
This particular tension in Hamlet's family has a profound impact on Ophelia and Polonius's family because it is this tension that is the principal impetus to the challenging words Hamlet eventually hurls at Ophelia. It is those words and Hamlet's unfathomable, altered behavior that lead Ophelia to madness and drowning. It is also the tension that promptts Polonius and Claudius to spy upon Hamlet, referred to when Claudius says, "we shall sift him":
CLAUDIUS: He tells me, my dear Gertrude, he hath found
The head and source of all your son's distemper.
QUEEN GERTRUDE: I doubt it is no other but the main;
His father's death, and our o'erhasty marriage.
KING CLAUDIUS: Well, we shall sift him.
In act 1 scene 2 there is considerable tension between Hamlet and both his mother, Gertrude, and his new step father, Claudius.
Hamlet has just returned home to find that his father, the king, has died and his mother has quickly married his uncle, the king's brother. Hamlet is understandably upset, but his mother tells him to, "cast thy nighted color off, And let thine eye look like a friend on Denmark" (I.ii.70-71). She is telling him to look at the new king, Claudius, as a friend. This is something that Hamlet is not ready to do. In his first soliloquy Hamlet is upset at his mother for her quick marriage, and he says, "Frailty thy name is woman" (I.ii.148). He believes that all women, including his mother, are weak. This sets up later feelings and conversations towards Ophelia, the other woman in the play. Hamlet is clearly disappointed in his mother, and she just wants him to accept the decision she has made. She should probably be more concerned with her grieving son, but she seems to be more concerned with Hamlet accepting Claudius.
Hamlet is also upset with Claudius, but Claudius does not help the situation when he speaks to Hamlet. Claudius tell him,"'Tis sweet and commendable in your nature, Hamlet, To give these mourning duties to your father; But you must know, your father lost a father; That father lost, lost his, and the survivor bound In filial obligation for some term To do obsequious sorrow. But to persever In obstinate condolement isa course Of impious stubbornness; 'tis unmanly grief" (I.ii.89-96). Claudius is telling Hamlet that death is a natural part of life, which it is, but he is also telling him that he is being stubborn for being upset and that it is unmanly to grieve his father's death for so long. This is one of the first conversations between the two that foreshadows the confrontation they will have throughout the play.