Let me start by recommending Paul Britten Austin's 1812 trilogy, which covers the entire saga of the Grande Armée and its opponents. Britten Austen adds relatively little of his own narrative and prefers to tell the story through the words of the people who were actually there. The last book in the series perfectly captures the misery of the retreat for the French soldiers and their allies (yes, Napoleon's Grande Armée also had Polish, German, and Italian soldiers who died a long way from home).

Between historical accounts and archaeological finds, there were four main causes of death for the soldiers and camp followers (including thousands of women): exposure (i.e. freezing), starvation, disease, and enemy action.

Soldiers in the 1812 retreat usually didn't die of just one thing. They might be so cold and exhausted, they couldn't defend themselves from the cossack who sabered them. Or they might be so malnourished and exhausted they couldn't fight off the typhus infection that finally killed them. The bodies found in mass graves show most men and women who died were sick, hungry, cold, and even wounded.

We do know it was quite cold during the October-December retreat to Vilnius. One of Napoleon's sergeants wrote in his diary abiut the cold:

On December 9th we started out for Wilna [Vilnius], with the temperature at -28 degrees Celsius [-18 Fahrenheit]. Out of two divisions totalling over 10,000 men...only 2,000 actually made it to Wilna. The hope of making it to the city where we were to find food in abundant supply had restored strength in me, or rather, like my fellow men, I produced supernatural efforts to make it there.

Thousands of the men who made the trek to Vlinius died after reaching their goal. Nearly 20,000 men died of diseases like typhus, starvation, exhaustion, and wounds in Vilnius. In 2001, Lithuanian archaeologists found over 3,269 bodies (including over two dozen women) in a mass grave in Vilnius.

According to researchers, some of the victims had frozen to death. Professor Olivier Dutour, who worked on the Vilnius graves project, said:

"The results confirm the extreme cold at this period. Napoleon claimed that he ordered the retreat because of the cold. And it's clear that some of the corpses really were frozen solid when they died in temperatures of -40C. It was like being in a deep freeze."

Some of the skeletons were found in the fetal position people take when freezing to death. Dutour suggests some soldiers curled up as they froze to death. Then, they were buried while still frozen solid.

Dental studies done by the team of University of Central Florida researcher Sammantha Holder and bioarchaeologist Dr. Tosha Dupras found evidence of disease and hunger.

High levels of nitrogen in the corpses' teeth showed that all the men were badly malnourished. Furthermore, over one-quarter of the dead had died from to epidemic typhus, a louse-borne disease. Typhus causes the body to lose water through urine, sweat, and diarrhea, which leads to a rise in nitrogen isotopes in the body.

Some of the men also showed signs of severe injuries in varying stages of healing, including one French soldier whose face had been slashed open by a sabre and had almost fully-healed before he died of typhus.

A 2006 dig in the town of Kaliningrad (formerly the East Prussian town of Königsberg), found over 800 bodies from Napoleon’s army. This cemetary had been next to a military hospital we know was treating typhus victims, so it's further evidence of the role of disease in killing off Napoleon's men.

Sources:

Mass Graves from the period of the Napoleonic Wars by A.N. Khokhlov and A.P. Buzhilova

"Digging Napoleon's Dead" by Jarrett A. Lobell

Adding a bit to the response below, it is worth noting, that the temperatures in Russia during the retreat from Moscow were definitely not mild and severe weather conditions substantially contributed to the number of casualties in the Napoleon's Great Army.

If we are to believe the quite famous chart made by Charles Joseph Minard who based his work on the contemporary sources, including diaries of commanding officers, such as Philippe Paul and Anatole de Montesqieu-Fezensac as well as diary of Pierre-Irénée Jacob, the army's pharmacist present during the retreat from Moscow, then the temperatures in that period were very low and could have been considered extreme by French and other Western Europeans accustomed to more temperate, maritime climate (on a side note, in the initial part of the campaign, French soldiers were bemoaning the extremely hot temperatures (reaching 35°C or 95°F), some of them even concluding that the conditions in Eqypt weren't as taxing as Russian summer was). According to Minard's data, the temperature fell to around 0°C (32°F) in late October, then oscillated between -11°C (12°F) and -25°C (-13°F) in the from 9th to 28th October and then dropped to -30°C (-22°F) on 1st of December with the lowest minimum at -36°C (-33°F) recorded on 6th of December. These numbers are corroborated by doctor Louis-Vivant Lagneau in his 'Journal d'un chirurgien de la grande armée' (Journal of the surgeon of the Great Army). Such temperatures, although maybe not impressive for Siberian winter's standards, are indeed very low for the late autumn in the central part of European Russia (just for comparison, there was only one day in the last quarter of 2018 when the temperature in Smolensk fell below -15°C (5°F) and record of lowest temperatures recorded between 15th November and 15th December 1941 shows only 3 days when the temperature fell below that point, with the lowest being -24.6°C or -12°F).

Now, it is worth stressing that Napoleon technically won the campaign by achieving all stated goals, i.e. defeating the Russian forces defending Smolensk and Moscow and capturing the cities, although this in no way changed the fact that the Russians were still defending. Furthermore, as the latter were controlling vast part of the country beyond the reach of invading troops, their situation was steady, while Napoleon's Army quickly began to suffer supply problems. Assumption that Russian will capitulate or at least begin negotiations after the capture of Smolensk and Moscow led Napoleon to believe that the hostilities will cease before winter and thus the invading army was illy equipped for the adverse weather conditions in late autumn. Soldiers were not issued winter equipment what heavily contributed to hypothermia and resulting problems during the retreat and also made it almost impossible to move artillery due to the lack of caulkined horseshoes (later, lack of provisions made horses too malnourished to be used, so large part of them ended up being slaughtered for food, essentially annihilating most of cavalry and artillery).

Additionally, during the retreat from Moscow, Napoleon's Great Army has taken part in several engagements, most notably the crucial battle at Maloyaroslavets, battles of Chasniki, Wyazma, Dorohobuzh, Smolany and Krasne, as well as the crossing of Berezina River while marauders and smaller units unable to follow up were constantly harassed by Russian light cavalry and partisans. Combat losses in these engagements were significant, as the combined losses in the Battles of Krasne and Berezina amounted to 50.000 soldiers (killed, taken prisoner or otherwise unaccounted for) what constituted roughly half of the total force at the beginning of the retreat. Of course, the malnutrition, hypothermia, pneumonia and typhoid fever certainly contributed to lowered combat prowess and morale that led soldiers to defeat or surrender but low temperature alone was generally not the primary direct cause of losses in the Russian campaign.

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Neuman, J., Flohn, H. 'Great Historical Events That Were Signiticantly Affected by the Weather: Part 8, Germany's War on the Soviet Union, 1941-45.' in: Bulletin of American Meteorological Society, Vol. 68, No. 6, June 1987.

Zamoyski, A., Moscow 1812: Napoleon's Fatal March, Harper Collins, 2005.