Claudius, as the most powerful (and the most evil) figure in the play, would be the character to look to for an example of abuses of power. But early in the play, Claudius, despite his ill-gotten gains, appears to be a competent and assertive monarch. In fact, one of the many complexities of the play is that Shakespeare did not choose to portray him as a murderous tyrant. One might argue that he abuses his power by using Polonius to spy on Hamlet, whom he has already named as his heir. But his first clear abuse of power is when he sends Hamlet with Rosencrantz and Gildenstern to England with a sealed diplomatic letter calling on England to execute his son-in-law. Similarly, his plot with Laertes to murder Hamlet at the staged bout in the final scene of the play represents an abuse of power. It is interesting, though, that Claudius, while gaining power using the most despicable means possible, does not abuse it until he feels threatened by Hamlet.
Yes there is an abuse of power! Claudius, the brother of the king, KILLS THE KING in order to become king himself. Then he MARRIES THE QUEEN! Um, ... if that's not an abuse of power, then I don't know what is!
Now, how do we know that this is the truth? In my opinion, the best scene to point to is when Claudius is admitting his guilt while praying and asking for forgiveness.
My fault is past. But, O, what form of prayer
Can serve my turn? 'Forgive me my foul murder?'
That cannot be; since I am still possess'd(55)
Of those effects for which I did the murder—
My crown, mine own ambition, and my queen.
Here we can see, point blank, Claudius admitting to the murder of the king. Claudius, of course, has no idea that Hamlet is listening here. Further, we can see the two reasons behind the murder. Claudius names these motives as "my crown" and "my queen" ... and he neatly adds "mine own ambition" which, in my opinion, puts him in league with Macbeth.
In conclusion, let's look at the irony here as Hamlet hears this confession, but fails to act.
A villain kills my father; and for that,
I, his sole son, do this same villain send
O, this is hire and salary, not revenge!
I suppose you can say that THIS is abuse of power, too. Hamlet halts avenging his father because of the confession of the murderer. In fact, don't even get me started on Hamlet, he abused his power of love over Ophelia and made her go insane. So, yes, the abuse of power is rampant in this play. As a result, everyone ends up dead.