My poetry is usually a way for me to play with language and gives me "license" to pun around (which I love to do -- cf. Liaison d'Etre). I like double meanings such as in the titles of Revere the Modern and Moving Violation. (I'm still trying to find a way to use "faux pa" as a phrase for stepfather.) I sometimes come up with a title and then build a poem or song around it (more commonly with my songs, since I write in song cycles and have certain ideas I want to fill in). Or one line just pops into my head from nowhere ("one if by phone, two if by fax" and "my poetic license had been revoked" are two examples) and I build on it. Or I know I want to say something about a certain subject (e.g., Sappho) and I work on that one idea.
To pursue one example, the entree to my poem on Sappho (Ancient Fire) was "sing me, muse" -- the line that starts both Homeric poems. Much of the density of this poem is an echo of Sapphic and other Greek verse (also dense), and derives in large measure from the many allusions and quotes here from ancient history and poetry: "she is like a god to me" is lifted from a Sappho poem, though I changed the sex from "he is like a god to me"; "art cut short when night grew long" is a reference to the triumph of Christianity and the start of the long night of the Dark Ages through the partial reversal of Horace's line "ars longa, vita brevis" (art is long, life is short); "whose Christian fate it was to burn" is a reference to the burning of Sappho's works by the early Christians; "your bright sister" is a reference to Plato's line about Sappho being the Tenth Muse; "who sang that song of love for what one loves" is a reference to another Sappho poem, in which she says "the most beautiful thing on earth is that which one loves". That's a lot to pack into three stanzas, I suppose!