No Avari ever appear in the stories of Middle-earth.
Actually, that's not strictly true.
In the true meaning of the word - those elves who refused the Great March, and thus never left Cuivienen - their only named characters appear in late writings discussing the departure. Here we learn that the Avari were made up of equal parts of the second and third kindreds (the Tatyar and Nelyar - among the Eldar, these kindreds became the Noldor and the Teleri), with two leaders: MorwŽ and NurwŽ. Even further back in time, we come across the tale of the original 144 elves who awakened at Cuivienen, including the named couples: Imin and IminyŽ, Tata and TatiŽ, Enel and EnelyŽ. Of course, their names literally translate as 'One', 'Two', 'Three' - so these may not be historical persons. If they are, any of the latter four could be Avari.
After the departure of the Eldar, the Avari lived in a very dark place. In part this was literal - the only natural light in Middle-earth was starlight - but in part it was the result of the war the Valar had waged against Melkor. Yet again, Middle-earth had been shattered and broken, and the Avari were now living in an ecosystem which was in a state of rapid change. Worse, as part of the war effort, Melkor had created the orcs - by corrupting some of the elves of Cuivienen. We know that some of these survived the war, and after the Valar retreated, taking Melkor with them, his creatures would have come crawling out. In those early days, the horror of the orcs would have been similar to that of zombies: they may well have been people the remaining Avari once knew and loved, now twisted and turned to evil.
And so, despite their mistrust of the Valar, the Avari slowly spread out from Cuivienen. We know there were at least six tribes, widely sundered, because we are told their names - six wildly different versions of the word 'Quendi'. Some of the Avari, indeed, made their way along the route of the Great March, where they joined up with the various stragglers: east of the Blue Mountains, they lived in the wild woods of Eriador and Rhovanion with the Nandor/Silvan elves (who later became the isolated populations in Mirkwood and Lorien, though at the time there was no distinction between the two forests), while others reached Beleriand and joined the Laiquendi of Ossiriand. Still mistrusting the Valar, very few of them would have lived in Doriath, with its Maiarin queen - and it is said that the Avarin Tatyar were extremely unfriendly to their kin, the Noldor, once they returned.
The Avari make one other major contribution to the history of Middle-earth: when Men awoke in Hildorien on the east coast, with the first rising of the sun, they learnt language from the Avari and the eastern Dwarves. It was from the Avari that Men first heard of the Valar, and Valinor - and as a result of their words, the three kindreds of the Edain began to seek the West. Without the Avari, there would have been no Beren, no Earendil - and no Aragorn.