How is nobility demonstrated in Shakespeare's Julius Caesar and by which characters is it demonstrated?

The concept of nobility and honor is represented through the honorable actions of Caesar, Antony, and Brutus.

Julius Caesar is considered too ambitious to be left alive.  Brutus has noble intentions when he decides to take part in the conspiracy to assassinate him.  Mark Antony even acknowledges his nobility in a speech after his death. 

Caesar is a noble character, except from the perspectives of the assassins.  According to Caesar and Antony, he was just doing what he thought was best for Rome when he went against Pompey.  It was a necessary act, and not an ambitious one. Antony says he was noble after seeing his body.

O, pardon me, thou bleeding piece of earth,
That I am meek and gentle with these butchers!
Thou art the ruins of the noblest man
That ever lived in the tide of times.
Woe to the hand that shed this costly blood! (Act 3, Scene 1) 

This is why Antony vows to avenge his death.  In this way, Antony was acting nobly too.  After all, Brutus and Cassius are the ones who killed the rightful leader of Rome.  For Antony, there is nothing more honorable than leading an army against them. In his mind, the proscriptions were also honorable.  The men who lost their lives or property were enemies, and they were at war. 

Brutus is very idealistic in his goals with regard to how the assassination will take place.  He does not want to do it secretly, because that would mean they were just common killers.  He wants the people to see that he and the others are liberating Rome.  He also does not want to have anyone else killed, even Antony. 

Let us be sacrificers, but not butchers, Caius.
We all stand up against the spirit of Caesar;
And in the spirit of men there is no blood:
O, that we then could come by Caesar's spirit,
And not dismember Caesar! (Act 2, Scene 1) 

When Brutus commits suicide, it is again a noble act by Roman standards.  Brutus and Cassius discuss how they do not want to be captives marched in triumph through Rome.  It is more honorable for them to just kill themselves.  Antony gives a speech over Brutus’s body in which he states that Brutus did not do what he did out of ambition or greed, but because he thought it was the right thing to do for the good of Rome. 

ANTONY

This was the noblest Roman of them all:
All the conspirators save only he
Did that they did in envy of great Caesar;
He only, in a general honest thought
And common good to all, made one of them. … (Act 5, Scene 5)

Antony and Octavius tried to continue as stewards of Rome for reasons they both thought were noble.  Of course, while they start out working together, they eventually come into conflict.  You can see the beginnings of it in the way they interact with each other.  This conflict is shown in the sequel, Antony and Cleopatra.