In this context, withal means something very close to with. Some context and a paraphrase would be helpful. Mercutio has challenged Tybalt to fight. Tybalt asks what Mercutio wants. Mercutio replies:
Good king of cats! All I want is one of your nine lives. I intend to make bold with it. Depending on how you respond to that, I may also beat your other eight lives out of you.
Make bold with means: act daringly with, or treat freely in any way I please, as though I have complete right over it.
So, why withal instead of with here? Because there is often (though not always) a shade of meaning in withal that isn't quite captured simply by with. The former has a sense of completeness that is absent in the latter: I see Rosencrantz with Guildenstern would simply mean I see the two guys together; I see Rosencrantz and Guildenstern withal would suggest that I'm thinking of the two are a complete unit, each part and parcel of a single whole, or each enhancing the other.
Let's explore some further examples. In The Merchant of Venice, Portia displays three caskets and instructs her suitor:
The one of them contains my picture, prince.
If you choose that, then I am yours withal. (II.vii.14–15)
Strictly speaking, withal can simply mean in addition: choose the correct casket, and you get me too. Denotatively, the speech says:
If you correctly choose the casket containing my picture, then you get me along with that.
But thanks to the shade of completeness or wholeness that is typically found in withal, connotatively, Portia suggests that that if her suitor chooses the right casket, she will by the same token be altogether his. Choosing the correct casket is inseparable from choosing Portia. They are the same thing.
Or, in Hamlet, the prince says to Horatio:
Horatio, thou art e’en as just a man
As e’er my conversation coped withal. (III.ii.56–57)
Horatio, you are as righteous a man as I have ever conversed with.
But the further idea is that out of all the people he has ever spoken to, Hamlet has never found anybody more reliable or upright than Horatio. There is a sense of completeness here: Hamlet is referencing his complete conversational history with any and every person.
So in Mercutio's speech, he says he will make free with Tybalt's life, but he is implying that he will make completely free with it, without holding back.