How is Virgil's Aeneid relative to other literatures of ancient times?

It'd be a bit much to say it's based on Homer, but it does certainly play on Homer an awful lot. And yes, it does play on plenty of other earlier poetry too: the biggest ones that we know of (remember only a tiny fraction of the poetry Vergil knew has survived to the present) would be Apollonius' Argonautica and Lucretius' De rerum natura.

Apollonius' epic is the only full-scale heroic epic other than Homer to have survived in Greek from before Vergil's time, and it played on Homer extensively itself. Sometimes, for example with some similes that appear in all three, you can see a development and shifting in emphasis from Homer to Apollonius to Vergil. (And if you want you can usually extend the progression still further, to Milton!) The Aeneid doesn't have so much in common with Apollonius in terms of their conception of what it means to be a hero, but the story of Jason and Medea in Argonautica 3 is the main intertext for Vergil's story of Aeneas and Dido in books 1 and 4; Dido's fury doesn't precisely mirror Apollonius, since Medea and Jason are still on good terms (ish) by the end of the epic, but it reflects the well-known outcome of the Medea story.

Lucretius' poem isn't a heroic epic but a philosophical one: it's a didactic piece on the teachings of Epicurean philosophy. However, Vergil made a lot of use of Lucretius for creating his own epic language and for a lot of the imagery that Lucretius uses: Lucretius has no hero and no battles, but he does have a lot of poetic imagery, similes, and analogies that he draws for the physical and not-so-physical phenomena he talks about. There's also quite a lot about the gods, even though the position of the gods in Epicurean philosophy is a big unknown area (Lucretius heavily emphasises the materialist nature of Epicureanism). Lucretius in turn also drew on older didactic poets, like Empedocles.

You also see incidental use of material from other poets: for example he alludes to Catullus a few times (one famous simile in Aeneid 9 is a revision of one of Catullus', and when Aeneas tries to defend himself to Dido's ghost in book 6, he misquotes Catullus). There's probably a lot of use of Ennius' Annals, but since only fragments of Ennius survive it's hard to pick them out.

Edit. Aaaaand I forgot to mention the Epic Cycle, even though I was planning to. We know the Epic Cycle as a set of early Greek epics that between them told the complete story of the Trojan War from beginning to end. It included the two Homeric epics, but the other six epics are all lost. By the 1st century BCE they were only really of antiquarian interest, but there was a resurgence of interest from some quarters in Vergil's time: that's when we see the Iliac tablets, artworks that draw heavily on the Cycle; Horace criticises the Cycle directly in his Art of poetry; probably the mythographic material that is our main source for the Cycle dates to around this time, or possibly a century earlier; and Vergil uses it for some elements of the Aeneid. He doesn't draw on it as a monolithic thing, though: the cyclic epics weren't designed to be a Cycle, but were originally independent poems.

Vergil's main targets appear to be the Sack of Ilion and the Aethiopis. We know depressingly little about the Sack of Ilion other than the fact that it related the destruction of Troy, so it's really just inference that Vergil's account of the sack in book 2 draws on the older poem, but it'd be hard to sustain an argument against any influence. We just don't know in which ways it influenced book 2. The Aethiopis is a bit easier because it had a more distinctive plot. Its influence is felt mainly in books 10-11: the Latin leader Camilla is pretty transparently modelled on the Amazon Penthesileia, the death of Pallas on the death of Antilochus, and Mezentius probably owes a lot to Memnon. There are a few other smaller allusions, but since the older epics are lost it's overwhelmingly likely that we simply can't spot all of them.

Now don't go reading Vergil as a derivative writer: in each case he puts his own very distinctive spin on his material. His skill in tying this poetic material into a single coherent poem earned him admiration, not accusations of plagiarism.