Recent advances in cancer research and diagnostics largely rely on new developments in microscopic or molecular profiling techniques, offering high levels of detail with respect to either spatial or molecular features, but usually not both. Here, we present an explainable machine-learning approach for the integrated profiling of morphological, molecular and clinical features from breast cancer histology. First, our approach allows for the robust detection of cancer cells and tumour-infiltrating lymphocytes in histological images, providing precise heatmap visualizations explaining the classifier decisions. Second, molecular features, including DNA methylation, gene expression, copy number variations, somatic mutations and proteins are predicted from histology. Molecular predictions reach balanced accuracies up to 78%, whereas accuracies of over 95% can be achieved for subgroups of patients. Finally, our explainable AI approach allows assessment of the link between morphological and molecular cancer properties. The resulting computational multiplex-histology analysis can help promote basic cancer research and precision medicine through an integrated diagnostic scoring of histological, clinical and molecular features.