It's interesting to note the wording of your question, which seems to suggest that Jane Campion's film Bright Star is less of a biopic on the life of John Keats and more of a adaption of Keats's single poem "Bright Star" (aka “Bright star, would I were stedfast as thou art”). While Campion’s film is a biopic, within her narrative of Keats’s entire life, we notice many links to the one individual poem from which the movie takes its names.
First off, there's the theme of love. Bright Star the movie focuses on the love between Keats and the seamstress Fanny Brawne. "A thing of beauty is a joy forever," states Brawne in the film. In "Bright Star" the poem, Keats confesses his desire to "feel for ever" the "soft fall and swell" of his "fair love's ripening breast."
In both works, we see love as an intoxicating, rapturous state. We also see themes of immortality. The extreme feelings of love last forever not because they literally last forever but because those moments feel so engulfing that they stay with you for as long you live.
One more link between the movie and the film is the theme of nature. In the poem, we have Keats envying the bright star for its ability to forever witness the beauty and majesty of nature. In the film, Campion shows Keats writing in trees or lying in a bed of flowers. Both the film and the poem attest to the enveloping and inspiring characteristics of nature.