We present a model for the formation of massive black holes (~1000 M ) due to stellar-dynamical processes in the first stellar clusters formed at early cosmic times (z ~ 10-20). These black holes are likely candidates as seeds for the supermassive black holes detected in quasars and nearby quiescent galaxies. The high redshift black hole seeds form as a result of multiple successive instabilities that occur in low metallicity (Z ~ 10-5 Z ) protogalaxies. We focus on relatively massive halos at high redshift (T vir > 104 K, z 10) after the very first stars in the universe have completed their evolution. This set of assumptions ensures that (1) atomic hydrogen cooling can contribute to the gas cooling process, (2) a UV field has been created by the first stars, and (3) the gas inside the halo has been mildly polluted by the first metals. The second condition implies that at low density H 2 is dissociated and does not contribute to cooling. The third condition sets a minimum threshold density for fragmentation, so that stars form efficiently only in the very inner core of the protogalaxy. Within this core, very compact stellar clusters form. The typical star cluster masses are of order 105 M and the typical half mass radii ~1 pc. A large fraction of these very dense clusters undergoes core collapse before stars are able to complete stellar evolution. Runaway star-star collisions eventually lead to the formation of a very massive star, leaving behind a massive black hole remnant. Clusters unstable to runaway collisions are always the first, less massive ones that form. As the metallicity of the universe increases, the critical density for fragmentation decreases and stars start to form in the entire protogalactic disk so that (1) accretion of gas in the center is no longer efficient and (2) the core collapse timescale increases. Typically, a fraction ~0.05 of protogalaxies at z ~ 10-20 form black hole seeds, with masses ~1000-2000 M , leading to a mass density in seeds of a few 102 M /Mpc-3. This density allows enough room for black hole growth by accretion during the quasar epoch.