By the end of Act I scene 3 of Macbeth, Macbeth decides not to kill Duncan and to wait and see what happens. What does this show about his character?

It is important to see the entire play as a process. In
this very important scene we see that Macbeth is completely taken back by the prophecy
of the witches and also the way that partly it has been instantly fulfilled by being
made the Thane of Cawdor. As he dwells on the truth of the prophecy with this regard, he
then thinks about the rest of the prophecy and if he will need to do anything himself to
make it happen. He finally concludes the following in an
aside:

If
Chance will have me King, why, Chance may crown me,

Without
my stir.

He basically thinks
that if it is decreed that he should be King, it will happen without him having to do
anything about it. This is important to note at this stage in the play, because it shows
Macbeth at the beginning of his downward plummet towards evil. He has logically
concluded that if it is prophesied that he will be King, then he doesn't have to act to
help destiny in its course. Waiting and seeing what "Chance" has for him shows that he
has not given himself over to evil by lending "Chance" a helping hand and killing Duncan
himself.