A premiere is an initial showing of a play, series, or movie, a premier is a head of government, and something that's premier is before any other in importance or order. All of this can be traced back to Old French premiere, meaning "first", through different routes. The noun premier came in 1711 from the adjective premier, because a premier is the most important minister in a country, and the adjective premier came sometime in the mid-1400s from a Middle French word with the same orthography and definition. The etymology of premiere meets up with those other terms in Old French because it also represents the first of something. That comes from Latin primarius, the root being primus, also "first". This makes premier a relative of the words principal, primitive, prince, prime, and many more. Through Proto-Italic priisemos, we can trace it back even further to Proto-Indo-European per, meaning "before", which sprung off hundreds of more relatives, some of which we've already covered.
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The Etymology Nerd