Your question is vague.
I find it useful to think about the process of characterizing a rubbery material as follows (I will assume no damage will occur in your material for your application):
Step 1: Choose a hyperelastic function. By definition this means that loading and unloading will be along the same path. Thus, finding the material constants for your hyperelastic function (e.x. initial shear modulus) will require that you perform tests (e.x. compression test) at an infinitesimally slow strain rate.
Step 2: Add on a viscoelastic function. Viscoelastic effects will cause the loading to stiffen and the unloading will no longer follow the loading path. You can find the material constants for your viscoelastic function by performing dynamic tests (e.x. Drop tests, DMA).
Step 3: Implement your completed material model into a finite element code. Now you can subject your material to various funky loads at various rates of loading and get good results -- now that your material model is fully characterized, as you say.
In reality, this is all quite complicated.
Thus, Jorgen has created a tool called MCalibration that allows you to choose your material model (i.e. choose your hyperelastic and viscoelastic functions) and then MCalibration will take whatever test data that you have and characterize your chosen model, automatically. Some will fit the data better than others. MCalibration also exports to some of the commercial codes.
Alternatively, you can use the material models that are built-in to your commercial code.
Depending on the kinds of loads (tension? shear?), strain magnitudes, and strain rates that you expect your whale blubber to see in its application will determine the material model that is most appropriate and the kinds of tests and the number of tests that you ought to perform. That information would help experts in the forum (not necessarily me) answer your original question, for example.
Sorry for the longish post, I was on a roll