Basically, everybody who signed the Treaty of Tilsit knew it was a sham from the start. Napoleon knew he didn't have the manpower to advance into Russia at that point in order to neutralize the threat from the east, although he had Austria and Prussia under his control at the time. Russia made a de jure declaration of war on the UK while actually not doing anything (and, in fact, continuing to trade with Britain.)

Napoleon, for his part, set about organizing the Grand Army. The Russians, on their side, implemented a number of reforms to beef up their own army, such as reducing the training schedule from 9 months to 6 months, and later to just over 3 months. Even if Napoleon didn't attack Russia, the Russians were intent on retaking Poland, at the very least. Britain was, simultaneously, engaging the French in Spain and lobbying throughout Europe for more allies. So much so that they were willing to burn down Copenhagen and steal the Danish fleet in 1807 just in case the Danes decided to side with the French.

In summary: The 1807 peace treaty was never meant to hold. Both sides agreed to it in order to rebuild and prepare for the coming war.

In short, Russia was refusing to follow Napoleon. France had set up the "Continental System" whose purpose was to hurt Great Britain economically, but Russia, egged on and supported by the British, refused to enforce the system. The purpose was to force Russia, which was not following the orders from Paris, to bend.

Napoleon also never actually saw Russia as an enemy, more as a recalcitrant ally, where Russia saw itself as in a battle for European hegemony with France. The plan was to make the Russians see the error in their ways and return to the fold. Napoleon was also apparently harboring ambitions to ally with the Russians, adding the forces together and going on to emulate Alexander the Great, conquering the Ottoman Empire and moving east to India.

Source: Moscow 1812: Napoleon's Fatal March by Adam Zamoyski