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":

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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":

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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.