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I am having a set of strings, they have been alligned and I have a matrix of pairwise distances. Now the objects have to be clustered yet I do not know the number of clusters beforehand. I would like to calculate some parameters that would help me decide - starting with intra-cluster variance and gap statistics.
The point is that I just have the distance matrix, not the coordinates themselves. What kind of analyses could I performed based on that?

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You can for example use the Ward's method implemented in scikit-learn or fastcluster. It will produce a dendrogram, and you scan spot at which level you should cut the dendrogram by looking at the variance decrease (sort of elbow method), cf. this post on se.

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