The Whites win in late-1919-early 1920, something that is very, very plausible. A junta is established between Denikin, Wrangel, Kolchak, and Yudenich. It is decided that Wrangel becomes head of government (probably alongside Milyukov and Pepelyayev) while Denikin and Yudenich are put in charge of the military. Kolchak becomes an élément grise in Russian politics, sort of like a Hindenburg-Ludendorff mix; a figure given deference who sort of stays behind the scenes but is the man to call when "decisive measures" must be taken to reestablish "order."
Meanwhile, the near-universally beloved Grand Duke Nikolai becomes Tsar, though this time he is a constitutional monarch with negligible power at best. This not only satisfies traditionalists and democrats, but also provides the nation with an untainted figurehead that pretty much everybody can rally around.
After an ill-fated Russian experiment with parliamentary democracy à la pre-fascist Italy; a theoretically liberal constitution ridden with institutional problems, a still-stratified society, instability, economic uncertainty, a political army. Kolchak (at the urging of Grand Duke Cyril who ascends the throne in 1929) finally steps in to reestablish "law and order" and becomes the de facto dictator of Russia. People who present a threat to his new order like Denikin and Kutepov and much of the Army high-command will probably be purged.
Kolchak probably dies some time in the '30s. During and after his reign, it's most likely that a movement like Aleksandr Kazem-Bek's Mladorossi gains momentum with its unique and popular (and typically fascist) slogans promoting both industrialization/modernization and traditionalism. Mladorossi the major (and eventually only) political party in Russia.
The charismatic and handsome Kazem-Bek becomes dictator after Kolchak kicks the bucket. How long his rule lasts is debatable due to his being of Azeri stock, but the major point is that his general philosophy (Tsar and the Soviets!) becomes the norm. His most likely successor is Anastasy Vonsyatsky (who may not be bigoted enough), maybe Konstantin Rodzayevsky.
Ethnicities that will most likely be targeted under the Whites (and I'm assuming we have pre-WWII Soviet borders here) are Jews (of course), Poles, Gypsies, Ukrainians (i.e. people who regard themselves as "Ukrainians" rather than "Little Russians"), Volga Germans, Crimean Karaites, possibly Chinese, perhaps Finns, and maybe Azeris depending on if and how far Kazem-Bek falls. Obviously homosexuals and non-Orthodox will be persecuted, the only exception being Muslims. Also expect a great deal of anti-Western sentiment.
Caucasians and Central Asian tribesmen like Kazakhs and Kalmyks and Buryats and Turkmen will in all likelihood not be targeted for two main reasons; the anti-Bolshevism that the majority of these groups displayed, and the distinct lack of general anti-Russian sentiment amongst them. In fact, the tribesmen will probably join the Cossacks in the pantheon of "martial races" (excellent theory, RGB). Again, the fate of Azeris largely depends on the success or failure of Kazem-Bek.
Interestingly, the concept of "sophisticated secret policemen vs. partisan street thugs" like we see exemplified in the rivalry between the SS and SA already existed in Imperial Russia. The "sophisticated" aspect—the Okhrana—will be reestablished, expanded, and revitalized under the leadership of people like Mikhail Diterikhs. The "street thug" element will lie in the resurrected Yellow Shirts under the command of somebody like Lev Okhotin or Gen. Vladimir Kozmin.
Another thing of note is the viability of Konstantin Rodzayevsky as the successor to basically any post. Head of the Okhrana, head of the Yellow Shirts, even Vozhd (or whatever the head honcho spot is called); Rodzayevsk could potentially succeed to any one (or perhaps even two) of these positions.
Basically, the regime we get is a cross between Iron Guard Romania and Francoist Spain alongside some major elements of clerical fascism.
Assuming that the Whites' borders are the same as OTL's pre-War Soviet Union, there will be lots and lots and lots of revanchist and imperialistic aims aimed at Poland, the Baltic States, and Finland. We also shouldn't discount potential adventures in Asia led by folks like Ungern fon Shternberg (if he's still around) or Semyonov.