How can pi have infinite number of digits and never repeat them? I am very confused about this matter, even if I searched google about this already. Please show me how this is determined and/or at least explain to me.
First, I saw this "Infinite Monkey Theorem" that says given infinite number of tries, a monkey could write a play of Shakespeare exactly. If so, why not "given infinite number of digits of pi, a pattern will form?"
Second, "if the digits are finite, then I can write this number as the ratio of two integers." And I found out that pi is the ratio of the circumference and diameter of a circle. If it this is so, then how come pi is proved to be infinite?

Assuming $\pi$ is normal (which seems likely, since most real numbers are normal, but is not proved), then yes, given enough digits of $\pi$, any pattern of finite length will be found.  However, in order for a number to be rational—that is, the ratio of two finite integers—such a pattern would have to be unending.  That is, either the digits of a rational number must be all zero after a certain point, or they must repeat forever after a certain point.  The normality of $\pi$ (if it is in fact normal) cannot guarantee such an eventuality; in fact, it forbids it.
As to your other question, yes, $\pi$ is the ratio of circumference to diameter.  If both of those were of integer length, then you would be right: $\pi$ would be rational.  However, the fact that $\pi$ is irrational means that either circumference or diameter can be of integer length, or neither, but never both at the same time.
A complete explanation of a proof that $\pi$ is indeed irrational is beyond the scope of this answer.  Further details can be found here: https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational
