Explain the predicament Brutus faces in Act 1 of Julius Caesar.

Brutus must decide how to act in response to the presumed
offering of a crown of rulership to Caesar.  The Roman Senate was a democratic body of
government, and Brutus and others are afraid that Caesar will be awarded special powers of solo
rulership.  He doesn't think that this is the correct form of government for
Rome.

So, his predicament is in how to respond.  Should he allow the
process to take its course and, if the people seem to want Caesar (a man that Brutus does admire)
as king, allow it to move forward; OR does he take sides against this potential change, siding
with the Conspirators, murdering Caesar to keep Rome a democracy ruled by the Senate? 
Assassination is an act of treason however, so, ironically, Brutus must decide whether to act
against Roman law in order to do what he thinks is best for Rome or obey the law and allow Rome
to become the sort of government he thinks is wrong.

Brutus, unlike
Cassius, is a cautious man.  In Act I, scene ii, when Cassius all but lays his cards on the
table, the only indication that Brutus gives of his discontent with the state of affairs is the
following:

Be not
deceived.  If I have veil'd my look,

I turn the trouble of my
countenance

Merely upon myself.  Vexed I
am

Of late with passions of some
difference,

Conceptions only proper to
myself,

Which give some soil perhaps to my
behaviours.

. . .poor Brutus, with himself at
war,

Forgets the shows of love to other
men.

It will be in the later Acts of
the play that Brutus' real feelings and the strain of his decision to murder Caesar is
demonstrated to the audience.