From a moral standpoint, one could argue that Shylock should not seek revenge because as a follower of Old Testament law he should leave vengeance or revenge to God.
However, from a human nature standpoint, Shylock has legitimate motivation for seeking revenge from Antonio (who also represents Christianity in Venice). Here are several offenses that Shylock has had to endure at the hands of Antonio and other "Christians."
1. As a Jew, he is segretated by Venetian law from the other city's residents.
2. Antonio has spit and cursed Shylock in public.
3. Antonio publicly criticizes Shylock, hurting his business.
4. A "Christian" steals away his only child (Jessica) and his jewels (his most prized possessions).
At the play's end, Shakespeare certainly demonstrates that seeking revenge is not only fruitless but also that it has destructive consequences, but most modern audience members feel sympathy for the broken Shylock in Act 4 and can identify with his reasons for wanting revenge.
Whether the character of Shylock is justified in "seeking revenge" on Antonio is a matter of perspective. William Shakespeare's The Merchant of Venice is replete with intimations of racial or ethnic hostility, as the Jewish moneylender is repeatedly on the receiving end of instances of anti-Semitic abuse. In the play's early scenes, it is clear that Shylock is deeply imbued with a sense of animosity toward Shakespeare's protagonists Antonio and Bassanio. The audience is not sensitized to the background of these relationships, so Shylocks early displays of hostility portray him in a particularly negative light. For example, take the following passages from Act I, Scene III. In the first, Shylock rejects Bassanio's invitation to join him and Antonio for dinner, prompting the following response:
Yes, to smell pork; to eat of the habitation which your prophet the Nazarite conjured the devil into. I will buy with you, sell with you, talk with you, walk with you, and so following, but I will not eat with you, drink with you, nor pray with you.
And, soon after, Bassanio introduces Shylock to Antonio, although it is clear that the two individual are already well-known to each other:
BASSANIOThis is Signior Antonio.
[Aside] How like a fawning publican he looks!
I hate him for he is a Christian,
But more for that in low simplicity
He lends out money gratis and brings down
The rate of usance here with us in Venice.
So it is established early on that Shylock loathes Antonio and Bassanio, but it is only with the following comments by Shylock that the audience begins to understand the roots of these sentiments:
Signior Antonio, many a time and oft
In the Rialto you have rated me
About my moneys and my usances:
Still have I borne it with a patient shrug,
For sufferance is the badge of all our tribe.
You call me misbeliever, cut-throat dog,
And spit upon my Jewish gaberdine,
And all for use of that which is mine own.
Well then, it now appears you need my help:
Go to, then; you come to me, and you say
'Shylock, we would have moneys:' you say so;
You, that did void your rheum upon my beard
And foot me as you spurn a stranger cur
Over your threshold: moneys is your suit
What should I say to you?
The relationship between Shylock and Antonio is one of serious mutual antipathy. Whether Shylock is justified in demanding the pound of flesh to which he is owed is, however, as noted earlier, a matter of perspective. Antonio entered into the arrangement with Shylock of his own free will. That the titular merchant of Venice did so out of his loyalty to his dear friend Bassanio is immaterial. A debt is a debt, no matter how heinous the terms. The barbarity of the debt is a direct reflection of Antonio's history of racism toward Shylock. Even from the standards of the period in which Shakespeare wrote his play, the penalty is extreme; by today's standard, the terms would never be permitted to be carried out. As such, from the perspective of the present era, Shylock would be justified in putting a lien on Antonio's business, but would not be justified in exacting mutilation.