What does the audience learn about Caesar’s character from Act II. sc.ii , and what finally convinces him to agree to go to the Capitol?

Caesar appears in very few scenes in this play that
happens to be called Julius Caesar, so it can be difficult to
arrive at a definitive description of his character.  The scene you mention, Act II,
scene ii, is the scene on the morning of his murder, when Caesar must choose between
listening to the prophesies and bad dreams of his wife that warn him to stay away from
the Capitol and ignoring these signs in order to go forward with his
plans.

In the beginning of the scene, Caesar does ask a
servant to:

Go
bid the priests do present sacrifice

And bring me their
opinions of success.

When
Calpurnia enters, begging him to remain home, Caesar speaks as if he considers himself
invincible.  He says that anything that might fright him, "when they shall see/The face
of Caesar, they are vanished."  He even goes so far as to say that "death, a necessary
end/Will come when it will come."

However, to heed the
augurers and appease Calpurnia, Caesar agrees to say he is not well and stay at home. 
So, though he appears to fancy himself invincible, in this moment we learn that he can
be swayed by his wife.

Finally, Decius Brutus, one of the
conspirators, arrives to convince Caesar to come to the Capitol, which he succeeds in
doing.  Decius Brutus flatters Caesar's vanity by saying that Calpurnia's dreams merely
foretell how important Caesar is to the future of Rome.  Caesar is satisfied with this
explanation and determines that, after all, he will go to the
Senate.

Throughout the scene Caesar seems very willing to
change his mind at the drop of a hat.  This seems to suit him just fine,
since:

. .
.danger knows full well

That Caesar is more dangerous than
he.

We are two lions litter'd in one
day,

And I the elder and more
terrible.

Caesar, it appears
from this scene, considers himself invincible, and so walks straight into his
doom.