After Godzilla's defeat in Godzilla '85 we find Tokyo trying to clean up after his little stroll through downtown and then there is a scene where some soldiers fight over some godzilla cells only to be gunned down by a foreign agent who then helps himself.
Cut to a scientist working with the Godzilla cells until terrorists blow up his lab killing his daughter, whose soul is transfered to a rose bush (don't ask!).
Then we jump ahead five years to the present (well 1989, but it was the present then!). Godzilla is returning and the authorities want the scientist's help to create a bacteria that will destroy nuclear material. They hope that this will stop Godzilla once and for all.
That wacky scientist! He combines Godzilla cells with the rose bush containing his daughter's soul to create a monster known as Biollante!
Now the terroists want that anti-nuclear bacteria and threaten to blow up the volcano containing Godzilla if the Japanese government doesn't turn it over to them. They agree but after the interference of a foreign agent Godzilla gets released anyway. Godzilla is strangely attracted to Biollante and heads off towards her. Time for a show down!