what is the relevance of computability when applying diagonallization?
There are no catches. Diagonalization is a very general proof technique that works in classical, constructive and computable setting. It is used to prove:

that there is no surjection from a set to its powerset
that the real numbers cannot be enumerated
that the computable real numbers cannot be computably enumerated
that the Halting oracle does not exist
etc.

In its most general form it is known as Lawvere's fixed-point theorem.
You ask what happens if you diagonalize against the rational numbers. You did not specify in what form the rationals are given, I am assuming you mean their decimal expansions. The diagonalization will produce a real number which is not a member of the enumeration (this real may be rational or irrational, depending on what enumeration you started with). If you start with a non-computable enumeration, the result may be non-computable. If you start with a computable enumeration, you will get a computable result, because diagonalization preserves computability.
Likewise, if you take a computable enumeration of computable real numbers (given as infinite sequences of digits), the result of diagonalization will be a computable real number which is not a member of the starting enumeration.