There is a customs union between Russia, Belarus, Kazakhstan, Kyrgyzstan as well as Armenia (the latter being not relevant to your case). This is how you proceed:
- When crossing the Russian, or the Belarusian, border by car, you will need to fill a "ТД-6" passenger customs declaration and fill your car details (VIN, make, etc.) thereinto. Search for "Russia PASSENGER CUSTOMS DECLARATION", or look it up here: http://eng.customs.ru/ . Print it out and fill by hand, preferably in two copies, as many border crossings will want to save paper on copying it and might force you to fill a second copy anyway. Notice: there is no need to fill the supplement if you do not carry cash or monetary instruments exceeding USD 10 000.
- Get a "green card" from your insurance company. You will need to present it at the customs control with your declaration copies.
- If your car title is 'electronic', get a paper copy from the relevant authority in your country. Paper title will be asked by the customs to prove you own the car.
- Tell the customs officer about your itinerary, Europe to Vladivostok, so they give you enough time for temporary importation of your car (up to six months, but exact dates are at the officer's discretion). Have proving documents ready if asked for them. Temporary importation may be prolonged for extra six months within the country, but will involve a lot of paperwork and queuing.
- You will get a stamp and a barcode sticker on your customs declaration (on your copy). Do not under any circumstances throw your declaration away or lose it. When leaving the EurAsEc customs union, you will need to stamp it out.
If crossing internal borders of customs union member states, always have your (original, stamped) customs declaration copy at hand. IDP is not required in Russia, if you have an EU driving license, but might be handy, especially in the 'more corrupt' countries.
Source: own experience, few times a year.