We cannot really prove weak dependence between two variables, but we can rule out different types of dependence, to a reasonable degree. One method is to use the chi-square test. This article does a decent job of explaining the process. https://stattrek.com/chi-square-test/independence.aspx

But a number of conditions need to be met. What kind of data are you working with? Also, apologies if the answer is overly simplistic. More details might help me provide a better one.

You can use regression to determine if there is weak dependence or show that the variables are normal in distribution

Regression assumes a lot about the type of relationship though.

The question is unclear. Does "I need to prove that my variables are weakly dependent" mean that you want to prove that the variables are:

(a) not independent;

(b) not strongly dependent.?

In the context of the central limit theorem,there needs to be some idea of an increasing data-set. What is this?

In some types of practical application, it may not be possible to "prove" anything in the sense of a mathematical proof starting from some standard model (that then itself needs to be justified), but you be able to justify (from the practical context and experience in that context) making the direct assumption that weak dependence applies. I hesitate to point you to my own publication Article Statistical Analysis of Empirical Models Fitted by Optimization

Again, when you say "I need to use the central limit theorem", are you wanting to use it to get a formal asymptotic distributional result without actually wanting to apply it , or do you need a practically useful approximate result? If the latter, it may be useful to concentrate on the variance, and its asymptotic behaviour, as this should give some clue about the effects of various levels of dependence (where, for studying the variance, interest is limited to the correlation properties only).