  We present a model for the formation of massive black holes ($\sim 1000
\msun$) due to stellar-dynamical processes in the first stellar clusters formed
at early cosmic times ($z\sim10-20$). The high redshift black hole seeds form
as a result of multiple successive instabilities that occur in low metallicity
$Z\sim 10^{-5}Z_\odot$) protogalaxies. We focus on relatively massive halos at
high redshift ($T_{\rm vir} > 10^4$ K, $z\gsim 10$) after the very first stars
in the Universe have completed their evolution. This set of assumptions ensures
that (i) atomic hydrogen cooling can contribute to the gas cooling process,
(ii) a UV field has been created by the first stars, and (iii) 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. A large
fraction of these very dense clusters undergo 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. Typically a fraction $\sim 0.05$ of protogalaxies
at $z\sim10-20$ form black hole seeds, with masses $\sim 1000-2000 \msun$,
leading to a mass density in seeds of a few $\simeq10^2\msun/{\rm Mpc}^{-3}$.
This density allows enough room for black hole growth by accretion during the
quasar epoch.
