In his first conversation with Brutus, Cassius makes him aware of just how feeble and human Caesar actually is. He clearly resents the general opinion in which Caesar is deemed a demigod and wishes to persuade Brutus that such a belief is a fallacy. In part, he relates the following encounter to prove his point:
... For once, upon a raw and gusty day,
The troubled Tiber chafing with her shores,
Caesar said to me 'Darest thou, Cassius, now
Leap in with me into this angry flood,
And swim to yonder point?' Upon the word,
Accoutred as I was, I plunged in
And bade him follow; so indeed he did.
The torrent roar'd, and we did buffet it
With lusty sinews, throwing it aside
And stemming it with hearts of controversy;
But ere we could arrive the point proposed,
Caesar cried 'Help me, Cassius, or I sink!'
I, as Aeneas, our great ancestor,
Did from the flames of Troy upon his shoulder
The old Anchises bear, so from the waves of Tiber
Did I the tired Caesar...
The implication is clear: Caesar was too weak to complete the swimming challenge and save himself from drowning. He relied on the much stronger Cassius to save him, which he did, like the heroic Trojan warrior, Aeneas.
... He had a fever when he was in Spain,
And when the fit was on him, I did mark
How he did shake: 'tis true, this god did shake;
His coward lips did from their colour fly,
And that same eye whose bend doth awe the world
Did lose his lustre: I did hear him groan:
Ay, and that tongue of his that bade the Romans
Mark him and write his speeches in their books,
Alas, it cried 'Give me some drink, Titinius,'
As a sick girl. Ye gods, it doth amaze me
A man of such a feeble temper should
So get the start of the majestic world
And bear the palm alone.
In addition, he informs Brutus about Caesar's other physical ailments. Not only is he weak, but he also suffers from what seems to be epilepsy. Cassius had seen him have a seizure and, as he had his fit, he turned pale and his eyes glazed. Caesar groaned because of his discomfort and requested a drink. Cassius cleverly contrasts the power that Caesar supposedly has, with the weakness he displays. He calls Caesar's lips 'coward', implying that Caesar himself is cowardly.
Furthermore, Caesar's steely-eyed command and his authoritarian look ebbs away once he is overcome by fainting spells. The tongue he uses to command others to write for him, becomes a feeble instrument begging for relief once he becomes ill. Cassius contends that Caesar has a poor constitution and he is amazed that such a frail individual should solely command such great authority and stature.
It is obvious that Cassius greatly resents the power that Caesar wields. The purpose of his conversation is to persuade Brutus to consider the fact that Caesar is not fit to govern. They can wield just as much power as Caesar does and are more equipped to lead Rome. It is thus imperative that they plot his overthrow. Brutus tells him that he will consider what Cassius has told him.
... What you have said
I will consider; what you have to say
I will with patience hear, and find a time
Both meet to hear and answer such high things.