The algebra of quasisymmetric functions QSym and the shuffle algebra of compositions Sh are isomorphic as graded Hopf algebras (in characteristic zero), and isomorphisms between them can be specified via shuffle bases of QSym. We use the notion of infinitesimal characters to characterize shuffle bases, and we establish a universal property for Sh in the category of connected graded Hopf algebras equipped with an infinitesimal character, analogous to the universal property of QSym as a combinatorial Hopf algebra described by Aguiar, Bergeron, and Sottile. We then use these results to give general constructions for quasisymmetric power sums, recovering four previous constructions from the literature, and study their properties.