We present a surface x-ray analysis of the atomic structure of the (0001) surface of the topological insulator Bi₂Se₃, which was grown as a single crystal and as an ultrathin film on Si(111) using molecular beam epitaxy (MBE). In general we find that the top Se-Bi layer spacing is expanded between 2 % and 17 % relative to the bulk, while deeper layers and the first van der Waals gap are unrelaxed. The top layer expansion is directly related to the amount of surface contamination by carbon and oxygen. The near-surface structures of the single crystal and the MBE-grown thin film differ in the degree of (static) disorder: for the former an overall Debye parameter (B) per quintuple layer (QL) of 5 Ų is found to decrease slowly with depth. MBE-grown Bi₂Se₃ films exhibit the opposite scenario, characterized by an increase in B from about 10 Ų for the topmost QL to values of B = 20-40 Ų for the fourth QL. This is attributed to the lattice misfit to the Si(111) surface. Ab initio calculations reveal carbon to act as an n-dopant, while the first interlayer spacing expansion induces a shift of the Dirac point towards the Bi₂Se₃ bulk conduction band minimum.