Is Antony motivated more by his hunger for power or by his desire to avenge Caesar's death? What textual evidence supports this?
Antony is motivated by both a desire for revenge and a hunger for power but Antony's actions after Caesar's death are motivated far more by a hunger for power than a quest for vengeance.
There's no doubt Antony had respect for Caesar and wants revenge for his death. At the end of the play, in Act V, 5 he calls Caesar "gentle" (a word of high praise) and declares "This was a man!" Earlier, he says to the dead body of Caesar in his first soliloquy in Act III, 1:
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 livèd in the tide of times.
Woe to the hand that shed this costly blood!
In other words, he does promise revenge: "Woe [sorrow] to the hand that shed this costly blood!"
But quickly after, in the same soliloquy, he turns to the larger political aspects of the death. Politics dominate his thoughts. His mourning takes on a quality that this is less personal than political, pointing to Antony's desire to have a strong voice in the public arena:
Blood and destruction shall be so in use,
And dreadful objects so familiar,
That mothers shall but smile when they behold
Their infants quartered with the hands of war,
All pity choked with custom of fell deeds
Rather than run from this vision of violence, in which mothers smile as their infants are murdered, Antony becomes a political player, jumping to action in his quest for power. A useful contrast is Hamlet, a play believed to have been written around the same time as Julius Caesar. Hamlet's quest to avenge his father's death is always personal and never political: he never has an interest in seizing the throne for himself, manipulating the public or raising an army. In contrast to Antony, he sees the danger in deception and deceptive words, whereas Antony jumps right in and embraces deception. In this way, he has more in common with the ambitious Macbeth and Lady Macbeth, who will do anything and deceive anyone to achieve power.
We see Antony as a man willing to manipulate and deceive to get ahead. He uses sarcasm in his speech to the plebians in repeatedly calling Brutus an "honorable man" in a way that exposes Brutus as dishonorable and incites the mob to "mutiny." And while there is no indication that Antony cries over Caesar's corpse, when he sees a servant do so in Act III, 1, he takes note of it:
Passion, I see, is catching, for mine eyes,
Seeing those beads of sorrow stand in thine,
Began to water.
When Antony gives his speech to the plebians in Act III, 2, inciting them against Brutus, he will weep: “his eyes are red as fire with weeping,” we learn, but the "red as fire" hints that it is more than sorrow that motivates him. Yet we are still left wondering if this could be the anger of one who simply wants to avenge his friend's death.
However, in Act IV, 1, the veils rip away and we see fully Antony's will to amass power. First, Antony, without a second thought, trades his nephew's life for the life of Lepidus's brother. He has no sooner done this, however, when he speaks of Lepidus (who is now offstage) with the most cutting contempt:
This is a slight, unmeritable man,
Meet to be sent on errands. Is it fit,
The threefold world divided, he should stand
One of the three to share it?
The last two lines reveal the breadth of Antony's naked ambition: he wants to divide the world three ways, and doesn't want to share it with someone he calls a "jackass" and a tool. When Octavius protests that Lepidus is an honorable soldier, Antony counters with the idea that his horse is also an honorable soldier. Like a horse, Lepidus "must be taught and trained and bid go forth." Antony shows his desire for power in this way: he wants to master and command other people, not be commanded. He says in Act 4, 1:
Do not talk of him [Lepidus]
But as a property.
Antony reveals his ambition as well when he says right afterward:
And now, Octavius,
Listen great things. Brutus and Cassius
Are levying powers. We must straight make head.
In other words, Brutus and Cassius are raising an army, so we must too. This is an open bid for power. Antony doesn't simply want to go into Brutus's tent at night and kill in him to avenge Caesar: Antony wants power, he wants kingdom, he wants to lead an army. Vengeance is a goal, but ambition is the overriding motivator of Antony's actions.