Cassius is not only fully committed to the conspiracy, but he is the instigator of it. The second scene in Act I reveals his character and his motives. Much of this scene is devoted to Cassius's attempt to bring Brutus into the conspiracy as its titular leader. Cassius knows that he is not personally well-liked or trusted but that Brutus has a sterling reputation with the Roman aristocracy and the common people, both for his own character and for his ancestors who were patriotic Romans and unselfish supporters of the republic. Cassius, according to Plutarch, talked to a great many Roman patricians about the danger Caesar represented, but Shakespeare did not have room to show Cassius intriguing with all of them. His conversation with Brutus in Act I, scene ii has to represent most of his efforts to instigate a conspiracy to assassinate Caesar. Cassius also invites Casca to dinner, obviously for the purpose of drawing him into this conspiracy, but what he says to Casca at that dinner is not included in the play. Once Cassius has won Brutus over to his scheme, he has no trouble enlisting others.

Caesar shows that he considers Cassius a threat. In Act I, scene ii he tells Antony:

CAESAR

Let me have men about me that are fat;
Sleek-headed men and such as sleep o' nights:
Yond Cassius has a lean and hungry look;
He thinks too much: such men are dangerous.

ANTONY

Fear him not, Caesar; he's not dangerous;
He is a noble Roman and well given.

CAESAR

Would he were fatter! But I fear him not:
Yet if my name were liable to fear,
I do not know the man I should avoid
So soon as that spare Cassius. He reads much;
He is a great observer and he looks
Quite through the deeds of men: he loves no plays,
As thou dost, Antony; he hears no music;
Seldom he smiles, and smiles in such a sort
As if he mock'd himself and scorn'd his spirit
That could be moved to smile at any thing.
Such men as he be never at heart's ease
Whiles they behold a greater than themselves,
And therefore are they very dangerous.

Cassius knows full well that Caesar dislikes him, and therefore he has several reasons for wanting Caesar dead. He is obviously envious and jealous of Caesar. He is afraid that the patricians will suffer under Caesar if he becomes king, because Caesar curries favor with the masses by giving them things he can only pay for by taxing those who have money and land. And furthermore, Cassius is in fear of his life. He suspects that Caesar would become a harsh ruler once he was elected king and that he could actually be put to death himself. At the end of Act I, scene ii, he says to himself:

Caesar doth bear me hard; but he loves Brutus:
If I were Brutus now and he were Cassius,
He should not humour me.

Cassius throughout the play is characterized as a greedy, selfish, cunning man. Another reason he has for wanting Caesar murdered is that he knows he could become a co-ruler in the new order, along with Brutus, and that he could make a fortune from graft in that position. Cassius is the real protagonist of the play. He senses that he could prosper if Caesar were eliminated but that he could suffer severely if Caesar became absolute ruler. He also knows that he has to have Brutus on his side in order to accomplish the assassination which is his original idea and his obsession. Without Cassius history would have been different, because Caesar would have survived and most likely become absolute ruler of the Roman Empire--which was exactly what he desired and was contriving to achieve.

In Julius Caesar by William Shakespeare, does Cassius give himself half-heartedly to this conspiracy, or does he commit himself fully?

Cassius instigates the plot to assassinate Caesar.  William Shakespeare’s drama The Tragedy of Julius Caesar is based on historical facts from 44 B.C.  In actuality for dramatic license, Shakespeare combines the number of senators to keep the play from being deluged with assassins.

There is no question that Cassius did not want Caesar to be the monarch of Rome holding supreme power.  Cassius was not alone; however, his reasons were more personal than some senators.  Brutus wanted Caesar to not rule Rome alone in fear that he would become too powerful. 

In several places, Cassius gives his reasons for wanting Caesar killed.  His testimony is clear, and there is no doubt that he was sure his cause was just.
 1st Statement

… but, for my single self,

I had as lief not be as live to be
In awe of such a thing as I myself.

Act 1, Scene ii—Cassius states that he does not know what other men feel but he would rather not live if someone who is his equal is so revered that Cassius would have to venerate him.

In the same scene:

When could they say till now that talk'd of Rome
That her wide walls encompass'd but one man?

Cassius sardonically states that there has never been only one man who could rule Rome because he was so great.  Caesar is no better than Cassius or Brutus.  In fact, Cassius calls Caesar womanly.  He is appalled that Caesar is so well-beloved that the senate would make him the ruler for life.

Cassius finalizes his conversation with Brutus with this statement:

Caesar's ambition shall be glanced at.
…
For we will shake him, or worse days endure.

Cassius lets the audience in this aside that the plot has already been formulated because he alerts Brutus that Caesar will be taken care of or Rome will suffer for it.

Later in Act I, Cassius along with Cinna meets Casca in the streets of Rome.  It is March 14, 44 B.C., and the day and night have had many portents and signs leading toward a significant event.  Cassius gives the interpretation that the Gods support his plan to assassinate Caesar through the storms and unusual events.   He also intends on meeting with the other assassins to formulate the final plans and then go to Brutus’ house to be sure that he will support the plot. 

There is no doubt that Cassius fully supported the assassination plan and was the primary leader.