Testing the CAPM is hard to do, generally, because it provides few readily falsifiable statements. For example the equation $$r_i-r_f=\beta(r_m-r_f)$$ can be rearranged to be $$y=\beta{x},$$ where $y=r_i-f_r$ and $x=r_m-r_f$. This implies that $\alpha=0$ in an estimation structured as $y=\beta{x}+\alpha$. The problem is that you will run into testing problems via Jeffreys' paradox.
If the CAPM is true, then Jeffreys' paradox holds strictly in this case. Jeffreys' paradox is a theorem which shows that all true sharp null hypotheses with data from a normal distribution will be falsified as the sample size becomes large enough. A sharp null hypothesis is any hypothesis of the form $\theta=k$. Because the dataset is so large, it would be stunning for it not to be falsified even if it is true. Conversely, if you consciously pared down your set to try to avoid the problem, then you will be unconvincing.
A good test and one that you can easily use to falsify the CAPM is to falsify Chebyshev's inequality. Because Chebyshev's inequality provides strict boundary conditions, if you regress on $$r_i-r_f=\beta(r_m-r_f),$$ holding $\alpha=0$ in the calculation, then I can tell you now that you will violate Chebyshev's inequality.
Assuming you have access to market data, then what you will do is calculate $$r_i-r_f=\beta(r_m-r_f)$$ with a calculation for $\hat{\beta}$ and holding $\alpha=0$. You will then calculate the residuals. The CAPM does not require that the data is normally distributed, only that a covariance matrix exists. That is still a pretty strong requirement as it reduces the possible distributions down quite a bit. Nonetheless, you cannot assume the empirical rule holds, but you can assume the Chebyshev's inequality holds.
If you find one observed residual outside the boundary permitted by Chebyshev's inequality, then there cannot possibly be a second moment in the underlying data. You will find plenty outside the boundaries. If there is no second moment, then it also follows that no covariance can exist. If there is no covariance, then the entire mental model upon which the CAPM is constructed collapses, although it also takes Fama-French regression with it too.
Testing with Chebyshev's inequality has the nice property in that even one observation rejects the null that the CAPM is true with perfect certainty. Indeed, you collapse Fama-French and the factor models and the APT too.
The library at Kenneth French's website won't help you. The Fama-French methodology was originally to test the CAPM. If the factors were not zero, then the CAPM is false. It doesn't make the Fama-French model true. Nonetheless, you can correctly make an argument under Frequentist Decision Theory, that if the null is falsified, then you should behave as if the alternative hypothesis is true until you find a better solution.
