What steps should I follow to build a model with such a microbiome dataset and what should I consider?

Cross-validation is a must for all classification tasks.

More specifically, to what extent should I account for within-subject correlation due to repeated measurements in the model?

All measurements of the same subject must be grouped. That means they are always part of the same fold during the cross-validation, be it for training or for validation purposes. They must be grouped as well if you are using them in unseen data.

Are these results biased?

Difficult to tell from the information you provided. If by evaluated each observation independently you mean that some measurements from identical subjects were used for training and some for validation, those results would be biased.