There are some accounts by journalists that cover Russia in this period: David Hoffman's The Oligarchs is a good option (it's in the AH book list), and there is also David Remnick's Resurrection, which is a followup to his Lenin's Tomb. Svetlana Alexievich's Secondhand Time covers the end of the Soviet period and is a bit more literary, but highly acclaimed.

It covers a broader span than just the 90s, but Stephen Kotkin's Armageddon Averted is a short history that covers the 1980-2000 period, and is unusual (in a refreshing way) in that he doesn't treat the pre- and post-1991 periods as entirely separate topics, but parts of one bigger historical continuity.

There are a few books that cover history in the other non-Russian Soviet states as well: Anatol Lieven's Baltic Revolution and Chechnya: Tombstone of Russian Power are good, if getting a little old (Anna Politkovskaya's A Small Corner of Hell also covers Chechnya). Olivier Roy has The New Central Asia is similar; Adeeb Khalid's Islam After Communism is excellent, and Phillip Hiro's Inside Central Asia is a broader, regional history. A history of the Caucasus that includes the post-Soviet era would be Charles King's Ghost of Freedom.

There is a whole literature by political scientists that analyzed the political aspects of the 1990s in Russia, and for whatever reason this hasn't really moved over into academic history yet. Richard Rose's How Russia Votes is now old, but a very interesting and in depth analysis of political trends in the early 1990s; same goes for Eugene Huskey's Presidential Power in Russia. Those are both pre-Putin, so they don't have that "endpoint" in their writing, which has its pluses and minuses.