2 answers
You can use Spearman's correlation, its a non-parametric measure of rank correlation. You should be able to get around the problem of different units. Another approach would be to use linear models to test associations instead of using correlations. Hope that helps.
I don't think you require data to be normally distributed for a linear regression
TCGA expression data is generally normalized, though you can also get the count data. Linear regression doesn't require your data to be normally distributed so I think you can use linear regression. If your uncomfortable using regressions you can use a non-parametric test like a Spearman's correlation which is independent of the nature of the distribution (given you have enough power).
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