The common (pearson) correlation coefficient measures linear association and can be influenced by a few influential points, but you can do rank (spearman) correlation, Kendal's Tau etc. which measure monotone association.

If data is skewed, you can transform the data to make it bell shaped and do whatever you want.

If two variables are nonlinearly related (by plotting them), you'd better do a nonlinear regression and assess the R^2 like measures to assess goodness of fit.