How did he portray it to the Confederation of the Rhine, the Prussians, the Austrians, etc. if not as something simply in furtherance of French interests?
Why did they join him? What was in it for them?
One aspect of the run-up to the crossing of the Niemen was that Napoleon portrayed the invasion as a crusade of civilized Europeans against semi-Asiatic despotism. The French Emperor ordered the press via the Ministry of Foreign Affairs to ramp up anti-Russian invective, stressing both the alienness of Russian political culture and people in comparison to civilized Europeans. This propaganda campaign crafted an image of a civilized West with France as its protector versus a despotic barbarism that was half-Asiatic.
This campaign did not invent this image of an alien Russian out of thin air. The Enlightenment philosophes often portrayed Russia as a demi-European state in which its rulers maintained the pretenses of Europeanness while they ruled via the knout. One aspect of Napoleon's propaganda drive was to translate and publish the English travelogue writer Edward Daniel Clarke's Travels in Various Countries of Europe, Asia and Africa which had very few kind words about Russia. Clarke portrayed Russia as a backwards and superstitious polity whose only real talent was in their imitation of more enlightened Europeans. This travelogue had already appeared on the continent in 1810, but Napoleon's propaganda machine translated it and disseminated it further throughout the French sphere of influence.
Napoleonic propaganda also invented some of their own anti-Russian screeds. The most infamous and long-lasting in public consciousness was the forgery, The Testament of Peter the Great. Likely written by Charles-Louis Lesur and appearing in the continent well after the invasion had begun, The Testament of Peter the Great alleged that Russia was following a consistent schema devised by Peter I to expand Russian power throughout the globe. The first stage of Russian conquest will be against both the Ottomans and Central Asia, which would then be followed by Russia encouraging a fratricidal war among Central European states. This divide and conquer strategy would leave Russia free to take both Istanbul and pry away pieces of Hungary and Germany with little costs. Like many historical forgeries, The Testament of Peter the Great greatly simplified complex events and mixed facts with conspiratorial fiction. The Testament would become in the nineteenth century what the Protocols of the Elders of Zion would in the twentieth, a screed that circulated among various intellectual circles that allegedly explained an existential threat to the West.
How much this propaganda had an impact on the Rheinbund and Italian states is difficult to gauge. General Sir Robert Wilson, the attache to Russian Army in 1813 noted that the French copies of The Testament were common among the detritus of the retreating Allied army in Vilna and that Maret's villa had numerous copies. But surviving letters from Rheinbund troops generally do not show much in the way of direct recapitulation of these stereotypes, but many troops did simply take it for granted that the Russians were an exotic other. Napoleon actually stepped up his anti-Russian propaganda effort in the aftermath of French defeat, stressing the barbarity of the Russians and the imminent terror of the Cossacks. This added to civilian fears of rape and pillage associated with the Cossacks. Yet such fears also coexisted with dislike of French rapacity and exploitation.
Compared to his propaganda effort, Napoleon's diplomatic initiatives to sell the war were somewhat anemic. He made overtures to Austria with the promises of new territory or even the return of Croatia, but made no firm guarantees. For Prussia, Napoleon gave the Hohenzollerns even less direct incentives for participation in the invasion. Again, there were low-level hints at a more favorable resettlement of territory, but nothing concrete. The Poles in the Duchy of Warsaw were likewise disappointed by Napoleon's lack of interest in elevating the Duchy into a restored kingdom. Napoleon had already been cooled of his ardor for Polish affairs and the elevation of Poland to a kingdom would have involved actually choosing a Polish king, which was no small affair. His great power allies in 1812 thus had no real political incentives to fight the war other than fear of Napoleon's wrath if they refused.
Napoleon in the period leading up to the invasion kept his diplomatic options open as to a final settlement of borders post-victory, but in doing so, French diplomacy became hostage to French military fortunes. In short, Napoleon's refusal to commit to any concrete political objectives encouraged even more dissatisfaction with French hegemony among his coalition partners. This rendered much of the Russophobic campaign nugatory for French interest. These anti-Russian ideas did have wide currency among European circles. For instance, Wilson would quote The Testament postwar as a warning for the British position in India. But this common cultural fear of the Russian Asiatic colossus was too narrow of base for Napoleon to build a coalition around, especially given the widespread unpopularity of French hegemony throughout Central and Southern Europe.
Sources
Hartley, Janet M., Paul Keenan, and D. C. B. Lieven. Russia and the Napoleonic Wars. New York: Palgrave, 2015.
James, Leighton S. Witnessing the Revolutionary and Napoleonic Wars in German Central Europe. New York: Palgrave, 2013.
Resis, Albert. "Russophobia and the" Testament" of Peter the Great, 1812-1980." Slavic Review 44, no. 4 (1985): 681-693.