Objectives and intentions and the whole Stanislavsky school of acting were created long after Shakespeare. So I'd argue that Shakespeare doesn't always write intentions: and here, Berowne is simply reflecting on Boyet, and thinking aloud.
That said, yes, he finds Boyet (who knows - but won't reveal - all the girls' secrets) extremely irritating. Berowne reflects on the fact that Boyet is extremely affected, though also extremely well-mannered:
This gallant pins the wenches on his sleeve;
Had he been Adam, he had tempted Eve;
A' can carve too, and lisp: why, this is he
That kiss'd his hand away in courtesy...
Berowne is also - rather painfully - aware of the fact that Boyet has access to the ladies all the time, and think he's great. So Berowne also itemises the qualities that might cause the ladies to like Boyet so much:
...the ladies call him sweet;
The stairs, as he treads on them, kiss his feet:
This is the flower that smiles on every one,
To show his teeth as white as whale's bone;
And consciences, that will not die in debt,
Pay him the due of honey-tongued Boyet.
And, not forgetting Boyet's slickness of speech - he is "honey-tongued". Hope that helps!
I agree with much of what robertwilliam said, however...
Stanislavski didn't "invent" objectives...we humans have them, and he created a vocabulary with which to discuss them.
I think the "key" in that speech, or the HOOK, if you will, is the line:
At some level, Berowne regrets his own lack of ability with ROMANTIC language (as opposed to his skill with rhetoric). This is futher evidenced by his interaction with Rosaline when he forswears "fancified" language.
So...his objective? I (as the actor playing Berowne) would start with something like this...
OBJECTIVE : "I want to ease my own heartsickness."
and build on that.