How does the author relate Chillingworth's and Dimmesdale's physical appearances to their mental, emotional, or moral states in The Scarlet Letter?

A good place to start in answering this question would be
Chapter Ten, entitled "The Leech and his Patient." This gives us a real insight into
both of these two characters and the way that their appearance is shaped by what is
going on within them.

Note how Chillingworth's eyes are
referred to in a frightening, almost supernatural way as he engages in his work as a
"miner":

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Sometimes, a light glimmered out of the
physician's eyes, burning blue and ominous, like the reflection of a furnace, or, let us
say, like one of those gleams of ghastly fire that darted from Bunyan's awful doorway in
the hillside and quivered on the pilgrim's face. The soil where this dark miner was
working had perchance shown indications that encouraged
him.

Clearly this description
of the "blue and ominous" light that is compared to the "ghastly fire" indicates how
Chillingworth is depicted as an almost demonic or evil character in his determination to
find out Dimmesdale's secret.

When we consider Arthur
Dimmesdale, you might want to think of the final paragraph in Chapter Nine in terms of
how the description we are given of him reflects his inner
anguish:

Alas!
to judge from the gloom and terror in the depths of the poor minister's eyes, the battle
was a sore one, and the victory anything but
secure.

Clearly, the inner
guilt that dominates Dimmesdale's "spiritual sickness" is impacting his physical
appearance and making him appear sick, pale and
distraught.

Thus both characters' physical descriptions are
shaped by what is going on with them internally. On the one hand, Chillingworth's
somewhat demonic appearance is justified by his obsession with "mining" the secrets out
of the interior of Dimmesdale's heart. On the other hand, Dimmesdale is clearly shaped
by his own inner guilt that is eating him up from the inside.