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I have entirely categorical data (survey results from users), so I've used k-modes clustering to better understand my users.
I'm not an expert at clustering methods at all. Is there a way to known way of estimating the importance of a feature (or combination of features) in deciding which cluster a user falls into?
It boils down to counting.
Compute which feature value has the highest probability of predicting a particular cluster. It's a straightforward application of Bayes' formula.
There is an amazing technique available for finding out impact of a different features on the model, it is called Permutation Importance.
To understand how PermuationImportance works please check my this answer on stackoverflow.
To see a working example with well explained code, please check this notebook.
Apart from this, if you want to learn how to manually analyse K-Means clustering algorithm please read this paper.
What about this: do some manual preprocessing first.
If you have many categorical variables, (Can't be that many for a survey.), for each one
Do so for each categorical variable. Sometimes it will be better to assign, say, only 3 major responses plus "other". Then do one-hot-encoding, (=categorical to numerical using dummy variables) then do simple KMeans clustering and interpret the resulting clusters yourself for plausibility.
If you have only free-text responses in your survey, or lots of NAs, you have to do even more preprocessing first.
Read about PCA("Principal Component Analysis") and while implementing must standardize data on similar scale.
