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He definitely used his own experiences with epilepsy and pre-epileptic visions when writing The Idiot, but that was much later. I think The Double had more purely literary antecedents. Doppelgängers were an established trope by then, which Dostoevsky got from the German romantics—people like E.T.A. Hoffmann, who specialized in stories with weird phenomena that could have a natural explanation (e.g. the mentally ill hero's delusional imagination) but are uncanny and probably magical all the same.

The Double was only Dostoevsky's second novel but it was one of the best things he ever wrote. It's fabulous! And short.

Poe also wrote a pretty disturbing short story about a doppelganger. https://en.wikipedia.org/wiki/William_Wilson_(short_story)

There is a good foreign film about a doppelganger called "The Double Life of Véronique" by the great European director Krzysztof Kieślowski. It's one of the few times I've watched a foreign movie more than once.