In the play Julius Caesar, what kind of a friendship do Brutus and Cassius have?

Brutus and Cassius have a complicated relationship where they are constantly manipulating each other.

Brutus and Cassius are not exactly friends.  They are both senators of the Patrician class.  They run in the same circles.  Cassius convinces Brutus to join the conspiracy against Caesar so that they can get more influential people to join and have the support of the Roman populace once Caesar is killed.

Brutus is definitely arrogant and principled.  Cassius is aware that he will need to appeal to Brutus’s ego in order to get him involved.

Men at some time are masters of their fates:
The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings.
Brutus and Caesar: what should be in that 'Caesar'?
Why should that name be sounded more than yours? (Act 1, Scene 2)

Cassius is careful to feel Brutus out, and make sure that he is worried about Caesar’s rise to power.  Then he tells Brutus that if they do not do something it is their own fault that they have to answer to Caesar, and goes a step further by telling Brutus that there is no reason he should not be in Caesar’s place.

Cassius’s appeal to Brutus’s name is no accident.  He knows that Brutus’s name lends considerable weight to the cause.  Yet Brutus is not comfortable just being a figurehead.  He takes charge immediately.  From the beginning, Brutus vetoes decisions that Cassius makes about who to include in the conspiracy and what to do with Antony.  Cassius wants to kill him, and to not let him speak at Caesar’s funeral.  Brutus overrules him on both occasions. 

Our course will seem too bloody, Caius Cassius,
To cut the head off and then hack the limbs,
Like wrath in death and envy afterwards;
For Antony is but a limb of Caesar:
Let us be sacrificers, but not butchers, Caius. (Act 2, Scene 1)

It is important to Brutus that the people see the conspirators as fair.  He does not want them to seem like murderers.  In his eyes, they are tyrant-killers.  He thinks that he will gain more favor with the people of Rome by allowing Mark Antony to speak.

These decisions are both disastrous.  Leaving Mark Antony alive was a big risk, because he did not turn out to be harmless.  Giving him permission to speak at Caesar’s funeral was even more dangerous.  Mark Antony was more persuasive than Brutus and turned the people against the conspirators, convincing them they were murderers.

Things get even worse when Brutus and Cassius are in the field.  Brutus has very little military experience compared to Cassius, and he is still in charge because he is the nominal leader of their army.  This leads to bickering and resentment between the men.  Cassius complains that Brutus punished one of his men, and Brutus accuses Cassius of rivalry.  However, their biggest argument is over going to Philippi.  Again, Brutus overruled Cassius, and they both paid the price.  They were stomped by Antony and Octavius's armies.

Brutus's insistence to be in charge and inability to take advice was dangerous to more than his friendship with Cassius.  After bringing Brutus in, Cassius had no choice but to take a backseat and let Brutus be in charge.  Although he may have meant well, Brutus had no idea what he was doing and led them into one disaster after another until they both ended up killing themselves rather than being taken in battle.