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The Hα spectral line is a well-studied absorption line revealing properties of the highly structured and dynamic solar chromosphere. Typical features with distinct spectral signatures in Hα include filaments and prominences, bright active-region plages, superpenumbrae around sunspots, surges, flares, Ellerman bombs, filigree, and mottles and rosettes, among others. This study is based on high-spectral resolution Hα spectra obtained with the echelle spectrograph of the Vacuum Tower Telescope (VTT) located at Observatorio del Teide (ODT), Tenerife, Spain. The t-distributed Stochastic Neighbor Embedding (t-SNE) is a machine learning algorithm, which is used for nonlinear dimensionality reduction. In this application, it projects Hα spectra onto a two-dimensional map, where it becomes possible to classify the spectra according to results of Cloud Model (CM) inversions. The CM parameters optical depth, Doppler width, line-of-sight velocity, and source function describe properties of the cloud material. Initial results of t-SNE indicate its strong discriminatory power to separate quiet-Sun and plage profiles from those that are suitable for CM inversions. In addition, a detailed study of various t-SNE parameters is conducted, the impact of seeing conditions on the classification is assessed, results for various types of input data are compared, and the identified clusters are linked to chromospheric features. Although t-SNE proves to be efficient in clustering high-dimensional data, human inference is required at each step to interpret the results. This exploratory study provides a framework and ideas on how to tailor a classification scheme towards specific spectral data and science questions.