Different distributions of charged-particles and leading charged-particle produced in proton-proton collisions at a centre-of-mass energy $\sqrt{s} = 13$ TeV are measured in the pseudorapidity range $|\eta| < 2.4$ for charged-particles with a transverse momentum $p_T~>~0.5$ GeV. Measurements are presented for four event categories. The first two categories correspond to inclusive and inelastic enhanced event samples. The other two categories are disjoint subsets of the inelastic enhanced event sample that are either enhanced or depleted in single diffractive dissociation events. The measurements are compared to predictions from Monte Carlo event generators which were tuned to describe the underlying event properties at lower centre-of-mass energies.