How does the metaphysical poetry of John Donne reflect his life and character?

This is an excellent question.  Donne was a very learned
man, and so the often highly intellectual nature of his poetry inevitably reflects his
deep and broad learning.  He was well read in numerous subjects, and so it is not
surprising that so many of his poems show such a wide range of intellectual interests. 
He was particularly interested in matters of religion, partly because he was raised in a
Catholic family during a time when it was technically illegal to be a Catholic.
Eventually, of course, he became one of the most highly respected priests in the
Protestant Anglican church. His religious poetry, therefore, often expresses a deeply
felt personal desire to think and speak truly about God.  "Satire III," for instance, is
a splendid example of Donne's very serious commitment to discovering truth in religious
matters.  At the same time, the poem also shows that Donne knew how difficult it could
be to arrive at such truth, especially during times of great religious controversy, such
as the era in which he lived.  It is possible to argue that Donne's deep interest in
religion affected nearly all his poems, including the ones that might seem at first to
have little to do with religion in any overt sense.  "The Flea," for instance, can seem
at first merely an erotic poem of secular love, but it is possible to argue that this
poem -- like much of Donne's apparently secular poetry -- reflects, through irony,
Donne's commitment to fundamental Christian ideals.  In other words, Donne may not
intend us to take the speakers of such poems as "The Flea" nearly as seriously as the
speakers take themselves. Such speakers, one can argue, are presented ironically in
order to teach spiritual lessons through clever indirection. Only a man as learned and
as well read and as interested in religious topics as Donne was could have written the
kind of poetry Donne composed. An excellent book on this topic is John Donne:
Conservative Revolutionary, by N. J. C. Andreasen.