We present a new study of the collisional runaway scenario to form an intermediate-mass black hole (IMBH, M-BH greater than or similar to 100M(circle dot)) at the centre of a young, compact stellar cluster. The first phase is the formation of a very dense central core of massive stars (M-* similar or equal to 30-120M(circle dot)) through mass segregation and gravothermal collapse. Previous work established the conditions for this to happen before the massive stars evolve off the main sequence ( MS). In this and a companion paper, we investigate the next stage by implementing direct collisions between stars. Using a Monte Carlo stellar dynamics code, we follow the core collapse and subsequent collisional phase in more than 100 models with varying cluster mass, size, and initial concentration. Collisions are treated either as ideal, 'sticky-sphere' mergers or using realistic prescriptions derived from 3D hydrodynamics computations. In all cases for which the core collapse happens in less than the MS lifetime of massive stars (similar or equal to 3 Myr), we obtain the growth of a single very massive star (VMS, M-* similar or equal to 400-4000M(circle dot)) through a runaway sequence of mergers. Mass loss from collisions, even for velocity dispersions as high as sigma(v) similar to 1000 km s(-1), does not prevent the runaway. The region of cluster parameter space leading to runaway is even more extended than predicted in previous work because, in clusters with sigma(v) > 300 km s(-1), collisions accelerate (and, in extreme cases, drive) core collapse. Although the VMS grows rapidly to greater than or similar to 1000M(circle dot) in models exhibiting runaway, we cannot predict accurately its final mass. This is because the termination of the runaway process must eventually be determined by a complex interplay between stellar dynamics, hydrodynamics, and the stellar evolution of the VMS. In the vast majority of cases, we find that the time between successive collisions becomes much shorter than the thermal time-scale of the VMS. Therefore, our assumption that all stars return quickly to the MS after a collision must eventually break down for the runaway product, and the stellar evolution of the VMS becomes very uncertain. For the same reason, the final fate of the VMS, including its possible collapse to an IMBH, remains unclear.