Despite the both being Allied Forces characters, there is little exploration of their relationship. However, they both go along well and are good friends. Russia is proven to see France as a person worth seeking advice from, while France seems to care about how Russia feels, often inquiring about how Russia seems happy after the gatherings of the Allied Forces.
In a relationship note between the main Allies, Russia is said to ask advice from France on how to make friends. He follows (stalks) France because he is shy to ask him directly. France finds it both terrifying and precious when he finds out where does the scary aura behind him comes from.When Russia and other countries went to France's home, Russia seemed to be really happy when staying at France's house. It made France happy knowing that Russia was enjoying it.
During the apendix time of Russia's and his European fellows' histories, Russians were really into French culture, to the extreme of Russia's boss ordering him to spy on France to see how he dresses. Russia was excited to know more about French culture and France happily obliged to teach him.