Calculating the information density of an encoding system

Question

How do I calculate the information density of an encoding system?

This question came up as I was reading several papers about encoding data in DNA (I think I have gone through just about every major paper on encoding data in DNA after Church's). All of them mention the information density of their encoding system, which I understand to be given in units along the lines of bits or bytes per nucleotide.

From looking around on the internet and asking around in the hbar (the Physics chat) I understand the absolute basics: that one creates a model that gives the probability distribution over the possible messages in your system (say, English, or ASCII) and then from there calculate the average information density of a message using that distribution.

Unfortunately, as a complete beginner, I don't know how to do any of those things. I'd like to understand that process and in the end be able to do my own calculations. Any resources would be helpful, of course, but I was advised to ask this question here.

Tl;dr: how do you calculate the information density of a system?