How should I measure correlation between, for example, population size or crossover probability and time ?

Pearson coefficient seems like an obvious choice. However my data do not follow normal distribution.

From what I remember Pearson should only be used when X and Y form bivariate normal distribution. Which is obviously not a case when X denominates mutation probability or population size.

Is there other way to measure correlation between variables or should I just use Pearson?

Furthermore how should I judge the strength of correlation? The tests I know about all rely on normal distribution in some way. Are there any others or should I just use rule of thumb?