William Wordsworth (51-53): In The Prelude (a long poem which William Wordsworth wrote with Coleridge in mind), Wordsworth says: "I did not pine like one in cities bred/ As was thy melancholy lot, dear friend" (VIII.433-34). This echoes Coleridge's lines (51-53) where he complains about his schooldays in London.
Plato and Neo-Platonism (whole poem): Plato and Neo-Platonism (a school of philosophy inspired by Plato, but developed later by a thinker named Plotinus) influenced the whole of Coleridge's poem, in a way. The idea that Nature expresses truths that reflect God, and that the Spirit finds a muted reflection of itself in the mind's thoughts, are all pretty Neo-Platonic.
German Transcendental Philosophy (whole poem): Coleridge's mystical musings also echo German philosophers like Kant, Schelling, and Fichte. Schelling was particularly interested in finding evidence of God in Nature, seeing the natural world as a divine revelation or "eternal language" (to quote Coleridge).
Hartley Coleridge (throughout): Coleridge's baby son, Hartley, is a big part of the poem. Coleridge hopes that he'll live a life of peace and joy close to God and Nature. But, to find out if he did, you'll need to read this brief biography (the short answer is that he actually had a lot of problems—like alcoholism—although he did manage to spend a lot of time in Nature).
Coleridge's School (Christ's Hospital, London) (25-44): In "Frost at Midnight," Coleridge complains about being bored in school at Christ's Hospital (not an actual hospital, just the name of the school) and being afraid of the brutal headmaster, James Boyer. Christ's Hospital actually still exists today, and it's apparently a pretty prestigious and old school (given how famous some of its alumni, like Coleridge, became).
Coleridge's "Stern Preceptor," Headmaster Boyer (38): His full name was James Boyer. Due to his position as headmaster at Christ's Hospital School, no one liked him. Well, it was partially for that reason, but more because he had a propensity for viciously thrashing students. Coleridge and the literary critic, Leigh Hunt, were pupils of his, and were both strongly opposed to Boyer.
Coleridge's Sister, Ann (43-44): In "Frost at Midnight" Coleridge remembers playing with his sister, Ann, as a child, when they were both "clothed alike"—not yet old enough to be distinguished from one another based on their genders. As a kid, he imagines that she might pay him a surprise visit while he's bored in school in London.