Macbeth alludes to Tarquin in act 1, scene 2 during his soliloquy as he sets out to murder King Duncan, who is asleep in his chamber. Macbeth states,

With Tarquin's ravishing strides, towards his design
Moves like a ghost.

Tarquin was the son of Tarquin the Proud, King of Rome, and is a chief character in Shakespeare's narrative poem "The Rape of Lucrece." These words allude to his actions when driven by lust. He enters Lucrece's chamber in the dead of night and rapes her, despite her desperate plea that he not harm her.

The reference is apt since it clarifies and accentuates the fact that Macbeth is also in the process of performing a similarly depraved and evil deed. Just like Tarquin, Macbeth has plotted his iniquity and, in his soliloquy, makes it quite clear that he understands the horror of what he is about to do. He is, however, so driven by his lust for power that nothing can stop him from seeing his wicked act through to its pernicious end, just as his desire for Lucrece directed the ill-fated Tarquin.

In the end, both Macbeth and Tarquin pay a heavy price for their turpitude. Macbeth loses his wife to suicide after being driven over the edge by guilt, and he is killed in battle by Macduff. Tarquin's entire family suffers banishment from Rome. The citizens revolt on discovering that Lucrece committed suicide because of the ignominy she had suffered at his hands.

Macbeth, in his soliloquy in act 2, scene 1, on the eve of murdering Duncan, states that "murder" (by which Macbeth means himself as murderer) is moving

With Tarquin's ravishing strides, towards his design.

The reference to Tarquin is a reference to early (quasi-legendary) Roman history, regarding the expulsion of the kings and the rise of the Republic. The last king of Rome, Lucius Tarquinius Superbus, was a notorious tyrant and (like Macbeth himself) utilized murder as a political tool in his rise to power. However, this specific reference (given the words "ravishing strides") actually seems to more specifically relate to Tarquin's son who raped Lucretia, a crime that would spur the overthrow the monarchy and the founding of the Republic.

A study of Macbeth shows the appropriateness of this allusion. Macbeth, as he references the Tarquins, is about to murder Duncan in order to seize the throne. With this reference, comparing himself to these notorious tyrants, Macbeth voices recognition to the severity of his crime. From this murder, he will seize the throne and plunge Scotland into tyranny.

The parallels are appropriate (particularly with Tarquin Superbus). Like Tarquin, Macbeth utilizes murder as a political tool, murdering not only Duncan, but also Banquo and the family of Macduff. Furthermore, in reaction to his crimes, he will face a rebellion aiming to restore Malcolm to his rightful throne.

In the end, like the Tarquins, Macbeth will be defeated. Where the Romans dismantled the monarchy and set up the Republic, Macbeth's opponents will restore the legitimate claimant, Malcolm, to the throne, reasserting legitimate kingship over Scotland.

There were several Tarquins. The reference to his "ravishing strides" suggests an allusion to Tarquin the Proud (Lucius Tarquinius Superbus). He was a tyranical ruler who was eventually overthrown (some say as a fallout from the rape of Lucretia, others for his arrogant and evil rule). Lucretia, like Duncan, was known for her virtue. Tarquin was attracted to her but because she was virtuous, the only way he could have her was through a violent rape that eventually cost her her life (she committed suicide). The story parallels in some way Macbeth's attack on virtue (Duncan) and establishes for the audience a suggestion that if Macbeth were to claim the throne he would reign as Tarquin did--with arrogance and tyrrany.

In this monologue, Macbeth is preparing to murder the king. Essentially, he is so worried about the act he is about to commit that his mind is playing tricks on him. First, he is unable to grasp the dagger, then he thinks he sees blood on the dagger; however, he comes to the conclusion that these manifestations are a result of his nervousness.
Prince Tarquin is known for raping the Roman matron, Lucrece; therefore, this allusion is very appropriate as it seems Macbeth is attempting to evoke the prince's tyrannical spirit in order to proceed with the murder.