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There are several techniques that you could apply in order to cluster data if your input is a matrix of pairwise distances between elements. As usual, the best option depends on your specific data, so it is hard to answer to the question of what is the best one, but you could try any of the following ones:


The k-medoids algorithm is similar to the well-known k-means algorithm. After randomly choosing k of your sequences as initial cluster centers (initial medoids) and assigning each sequence to the closest medoid, you randomly reassign sequences to different clusters as long as the value of the cost function decreases.
Hierarchical clustering is another example of clustering algorithm whose input is a matrix of pairwise distances between sequences. In this case the output is a dendrogram.
Another option is to apply multidimensional scaling, a dimensionality reduction technique which input is a matrix of pairwise distances between sequences, to project your sequences into a 2D plane. Once you do that, you can apply any cluster algorithm you can think of, like for instance k-means. 


As I said, there are many other options, but these ones are the simplest ones I can think of, and the ones I would start from.