The first "Best of the Best" never got a chance at the British box office. However, as a video release it did quite well from what I recall. Sadly this reputation was short-lived in Britain by its highly criticised sequel, which did make it to the big screen.
The 90's were a very cynical time and at the time of Best of Best's release martial arts cinema was in its the kickboxer-craze. In retrospect this was one of the worst times as far style in the genre goes, as kickboxing does not possess the asthetic quality of most styles. This was where Best of the Best stood out and at the time was probably my favourite film bar none. It provided good characterisation, good actors such as Eric Roberts, James Earl Jones, Sally Kirkland, Christopher Penn et al and good martial artists. The dominant style in the film is Taekwondo, performed by great Korean exponents of the style. Phillip Rhee, who was involved in the production of the film, had a solid background in both Taekwondo and Hapkido (the latter you don't see displayed until Best of the Best 2) and plays off superbly against Simon Rhee in the film's climax.
No need to go into plot details here as you will have got the idea from the other reviewers. The film should be credited for making some very obvious breaks from the action mould at the time. There is a strong attempt at a storyline, which, although a little cheesey by today's standards, had a different type of ending to most martial arts films of its day. The "sensei" of the film is a strong female character!
Before Marc Dacascos showed the western world that it was possible to have a great martial artist and serious actor in the same package, Best of the Best lead the way. The sequel, although inferior, is also worth checking for a change in direction of the "Bloodsport" style films.