Cesare Borgia didn't really "conquer" Milan.
King Louis XII of France, with the subtle but undeniable cajoling from Venice^1 , invaded the Duchy of Milan in 1499.
Louis, who had headed the army of his predecessor, Charles VIII, in his first intervention in Italy some five years prior, knew how volatile Italian politics could be^2 . He signed an alliance with the Republic of Venice, promising them half of Lombardy. The Support of the Papal State was guaranteed by offering support to a campaign in Romagna, to subdue the lords of the feuds of Camerino, Faenza, Forlì, Imola, Pesaro and Urbino; which were nominally vassals of the Pope but had ruled in near autonomy.
In June 1499, Louis appointed Giacomo Trivulzio^3 at the head of his army, and advanced from the west, while the Venetian army moved in from the east. Faced with a two-pronged attack, Lodovico fled to Innsbruck, where Maximilian, Archduke of Austria and Holy Roman Emperor (and historic enemy of France), made sure he was warmly received.
Truvulzo entered the city on the 6th of October without a fight, and King Charles appointed him governor. Cesare Borgia was indeed at his side when he entered the city, but his actual role in the conquest was minimal, and he was not made a cabinet member during the French rule; nor did he want to be. His goal was to secure men and resources from the French in order to embark on his campaign in Romagna. By November, having rounded up an adequate amount of resources men, he wrote to his father, who promptly disempowered all Papal vassals in Romagna; Cesare subsequently marched south to enforce this edict.
In all, Cesare Borgia's stay in Milan amounted to about a month. Thus, he did very little.
Footnotes:
The job was also made easier by the fact that Luois also had a score to settle with Ludovico Sforza, but that's another story.
In 1495, Charles had marched down to lay his vague claims to Naples unopposed at the invitation of Duke Lodovico Sforza (who had a weaker claim to the Duchy of Milan than the king of Naples, Alfonso II) and Pope Alexander VI (Alessandro Borgia, who had a personal spat with Alfonso). However, seeing the French army plough through Italy unopposed caused them to have a change of heart (along with the fact that Charles, like Alfonso, had a claim to Milan), and form a coalition with the Republic of Venice, The Republic of Genoa and the Duchy of Mantua: they met the French at Fornovo as they returned to Milan, and the result was an indecisive battle: the French broke through the Italian line, but they proceeded to go straight back to France. This betrayal led Charles to ask for and grant concrete guarantees from his allies; so that they would have something to lose if they again betrayed him. This betrayal also kicked off Charles' personal dislike of Lodovico.
Previously a general in the employ of the Kingdom of Naples, Trivulzio had organized the heroic Neapolitan fighting retreat during Charles' advance on Naples. Charles on reaching the city found it open to him; the King had fled to Ischia. Negotiations on his entry were held by Truvulzio. Charles was so impressed by him that he offered him a stipend of 10,000 yearly ducats to become Marshal of France. Trivulzio graciously accepted.
Conversations of the Italian wars are often dominated by stories about the Borgias, ignoring the complex dynamics at play, as well as a wealth of other, often more interesting, individuals, events and achievements vital to the time period. My favorite account of the Italian wars of given by good old J. J. Norwich, A History of Venice Vintage Books, New York, 1989. ISBN 0-679-72197-5 However, the account is very one-sided. Although he's outdates, a cool starting point is Charles Oman, A History of the Art of War in the Sixteenth Century Methuen, London, 1937.
The definitive account of the Italian Wars has been given by Prof. Piero Pieri Il Rinascimento e la Crisi Militare Italiana Einaudi, Torino 1970. However, I do not know if he has been translated into english. There was also some interesting research by the Smithsonian done about the same topic: Thomas F. Arnold The Renaissance at War; Smithsonian History of Warfare Collins, New York, 2006. ISBN 0-06-089195