In Julius Caesar, how has Brutus been feeling at the beginning of the play and how does this open the door for Cassius to draw him into his plot?

Cassius is the first to notice that Brutus is in a perturbed state and comments on his mood in Act 1, scene 2:

Brutus, I do observe you now of late:
I have not from your eyes that gentleness
And show of love as I was wont to have:
You bear too stubborn and too strange a hand
Over your friend that loves you.

Brutus replies:

Cassius,
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 behaviors;...

...Nor construe any further my neglect,
Than that poor Brutus, with himself at war,
Forgets the shows of love to other men.

Cassius believes that Brutus has an issue with him but Brutus assures him that although he is troubled, it is an issue that he has to resolve within himself. He is, in essence, in a quarrel with himself and has, therefore, forgotten to show courtesy to others.

Cassius perceives that Brutus is disturbed and thus easy fodder for his manipulation. He apologises for being wrong and immediately starts complimenting him, softening him up. He, for example, mentions that many esteemed Romans except Caesar have wished that Brutus could see himself through their eyes, suggesting that they see much good in him, something that he can't quite seem to quite notice himself. 

It is apparent that Cassius wants to paint a negative image of Caesar early in their conversation because he wishes to later ask Brutus to join his conspiracy. He mocks the general by referring to him as 'immortal' and saying that Caesar is not one of those who see the good in Brutus. Brutus, though, is aware of what Cassius is attempting and asks him:

Into what dangers would you lead me, Cassius,
That you would have me seek into myself
For that which is not in me

Cassius then mentions that he will hold a mirror to Brutus and inform him about his inner goodness and that if Brutus should know him as a mere flatterer who later turns against those whom he complimented, then he should see him as dangerous. He is clearly attempting to manipulate Brutus.  

When they hear shouting and Brutus expresses fear that the populace wishes to choose Caesar as king, Cassius pounces and asks if Brutus fears it and he suggests that, therefore, he does not welcome it. Brutus agrees and wants to know what Cassius wants of him. He asks if what Cassius is getting at is for the general good then he could offer him both honour and death and he would be indifferent to both. He further mentions that he loves honour more than he fears death. This means that he is prepared to sacrifice his life for the honour and good of Rome.

Cassius knows that Brutus is ready for his suggestion of a plot and he goes into a long discussion about Caesar's weaknesses, contrasting his frailty with their strengths. He expresses fear that Caesar might abuse his power if he should gain more status. He ends his entreaty with another flattering comment about Brutus:

O, you and I have heard our fathers say,
There was a Brutus once that would have brook'd
The eternal devil to keep his state in Rome
As easily as a king.

Brutus retorts by saying that he will consider what Cassius has said, but asks him not too work himself up too much. He gives him the following instruction:

...my noble friend, chew upon this:
Brutus had rather be a villager
Than to repute himself a son of Rome
Under these hard conditions as this time
Is like to lay upon us.

Brutus is evidently not prepared to allow Rome to fall under 'hard conditions,' as he calls it, that may ensue if Caesar should become its emperor. Thus the scene is laid for the conspiracy to progress and plot its bloody mischief.