How can amino acids sequences be used to determine the degree of relatedness between organisms?

By quantifying the amount of difference in the DNA amino acids they can create a evolutionary tree called a phylogeny, which groups organisms with the least amount of difference together. This is because evolution is though to proceed on the path of least resistance (although not allways).

Example: Suppose we identified the same gene in 4 different animals. and we found out the sequences listed below.

Organism 1: AAGGCGTAA

Organism 2: AAGGCGAAA

Organism 3: ATCGCGGAA

Organism 4: AAGGCGTAA

By examining these 4 organisms we can see that 1 and 2 are the same.

We put them together like this. (the horizontal lines are for formatting please ignore them, the important lines are in bold)

-/\

1--4

We see that Organism 2 is slightly different then 1 and 2 by one letter

so we infer it is only slight less related.

--/\

-/\-\

1-4--2

The 3rd organism is very different and is inferred to have diverged from the other organisms a long time ago. So we draw it as the farthest related like this.

---/\

--/\-\

-/\-\-\

1-4-2--3

Now I have made a phylogeny, and the inferred evolutionary history.

This Wikipedia article below will help iron out the rest of your questions.