When and to whom does Cladius openly reveal his determination to remain king? Quote specific passages to illustrate your points.

Claudius also reveals his determination to remain King when he says in Act III, scene iii:

But, O, what form of prayer
Can serve my turn? 'Forgive me my foul murder'?
That cannot be; since I am still possess'd
Of those effects for which I did the murder,
My crown, mine own ambition and my queen.
May one be pardon'd and retain the offence?
In the corrupted currents of this world
Offence's gilded hand may shove by justice,
And oft 'tis seen the wicked prize itself
Buys out the law: but 'tis not so above;
There is no shuffling, there the action lies
In his true nature; and we ourselves compell'd,
Even to the teeth and forehead of our faults,
To give in evidence. What then? what rests?
Try what repentance can: what can it not?
Yet what can it when one can not repent?
O wretched state! O bosom black as death!
O limed soul, that, struggling to be free,
Art more engaged! Help, angels! Make assay!
Bow, stubborn knees; and, heart with strings of steel,
Be soft as sinews of the newborn babe!
All may be well.

Although this is a long quote, it reveals Claudius' reluctance to repent because it would mean giving up what he has gained: his crown and queen.  Clearly he is not willing which he realizes results in condemnation.  He does not have the moral capability to do what he knows is right. He is willing to trade his eternal salvation to remain King and married to Gertrude.