  We introduce Macdonald characters and use algebraic properties of Macdonald
polynomials to study them. As a result, we produce several formulas for
Macdonald characters, which are generalizations of those obtained by Gorin and
Panova in [Ann. Probab. 43 (2015), 3052-3132, arXiv:1301.0634], and are
expected to provide tools for the study of statistical mechanical models,
representation theory and random matrices. As first application of our
formulas, we characterize the boundary of the $(q,t)$-deformation of the
Gelfand-Tsetlin graph when $t=q^{\theta}$ and $\theta$ is a positive integer.
