He was a fairly well-known artist internationally for the time period. Granted, given literacy and income levels, if you asked the average person in Italy (unless they lived in the town of Vinci) they probably wouldn't have a clear idea, if at all, as to who he was.
Leonardo's big break came when he moved from Florence to Milan. Leonardo, especially during his early career, can be described with the old trope, "Smart But Lazy." There's a weird four-year "blank period" immediately following his breakthrough successes in the 1470's in which there isn't a trace of his work. When he starts painting again, his bad work ethic gets even worse; he starts some works without finishing them, and even takes commissions without even starting them.
Somehow, Leonardo entered the circle of Lorenzo de Medici, who was the behind-the-scened ruler of the Florentine Republic (possibly Leonardo's dad, a very successful notary, introduced them). But Lorenzo's commissions did little to alleviate Leonardo's evident boredom, and he started drafting small-scale architecture and military machinery as he neglected his artwork.
In 1482 Leonardo da Vinci traveled to Milan with a golden lyre officially to take part in a music contest at the Milanese court, but unofficially to deliver a nine-paragraph letter to Ludovico Sforza. You see, Lorenzo's diplomatic modus operandi was to contrive, outside of the normal appointment of ambassadors, the employment of Florentine artists (normally painters) on projects commissioned by his allies. King Ferdinand of Naples, for example, had two Florentine painters working for him, while a group of painters had also been dispatched to Rome. This tactic got Lorenzo in everyone's good books, plus when he wrote to check up on how his dear artist friends were doing he could, by extension, find out what was going on in rival courts.
Although Leonardo's initial welcome was tepid, Lorenzo knew what he was doing, and Da Vinci adapted quickly; at first his rural Tuscan accent was a barrier to making headway in the Milanese court, but soon even his personal writings would be peppered with Lombard-isms. Leonardo had been awkward and passè in the refined Florentine high society (he would bitterly describe himself a "man without letters" both figuratively and literally; his handwriting was so bad he asked a scribe to transcribe his letters to Ludovico) while in the Milanese court his military imagination was fueled by the veterans of Duke Galeazzo Maria's near constant campaigning and the uneasy peace over which his successor Ludovico presided. Indeed, Milan had been bullying other cities in Italy for some two hundred years and this was the prefect environment for an aspiring military engineer. The ruling House of Sforza themselves were a cadet branch of the House of Visconti who had distinguished themselves as military commanders before Francesco Sforza came to the throne by marrying Duke Filippo Maria Visconti's only child and daughter, Bianca Maria.
Da Vinci's permanence in Milan is instrumental to his international recognition; the Duchy of Milan was one of the most internationalized states in Italy. Ludovico's predecessor as ruler, Galeazzo Maria Sforza, was an an unabashed francophile. Prior to ascending to the duchy, he had fought in the War of the Public Weal between the Duke of Burgundy and the King of France, commanding a Milanese contingent fighting in support of the King. Galeazzo Maria had also married Bona of Savoy, a native french speaker raised in Paris (against the advice of his mother) and maintained a large embassy at the French court, headed by his brother and eventual successor Ludovico.
Ludovico, who was precariously poised at the head of a regency for his spoiled and childish nephew, was very active on the foreign policy front, especially as his brother had lost Bergamo and Brescia to the Venetians; and when he came to the throne, apart from the Lombard heartland, the once massive Duchy of Milan could only count on a tenuous hold on two client-states: Genoa and Pisa.
At first Ludovico hoped to maintain his position through an alliance with the Kingdom of Naples; and he sent armed contingents to help Naples in a conflict against Ferrara (1484) and unruly barons (1486) while in 1488 he participated in a sumptuous wedding ceremony between his charge, Gian Galeazzo and Isabella of Aragon, in Naples. However, this friendship would not last.
Although young Gian Galeazzo was completely disinterested in ruling Lombardy (once overage he had every legal prerogative to dismiss his uncle, but he didn't) his wife Isabella of Aragon had different ideas, on arriving in Lombardy from Naples she immediately would not stand for the fact that she and her husband were to entertain a frivolous court in Pavia while Ludovico ran the country in Milan. Ludovico, for his part, was already looking beyond Italy to secure his borders; he maintained a guarded friendship with France (guarded because the suspicion prevailed that his brother was murdered on the orders of King Louis XI) and in 1494 married off his nice Bianca Maria Sforza to the Archduke of Austria, Maximilian von Hapsburg, who had been elected Holy Roman Emperor the year before, securing a vital ally immediately to the north of the Republic of Venice, Milan's historic rival in Northern Italy.
Bianca Maria was escorted to Innsbruck in a procession through the Lombard Alps into Austria with a massive dowry and enormous retinue, which according to some included Leonardo Da Vinci; in his Codex Atlanticus he records his thoughts on alpine geography, wines grown in the Alps, and hot springs. So there is a fair chance he interacted with the Austrian court, where we can say at least some people would know who he was.
On the other hand, we can be sure that the French were absolutely crazy for Leonardo; twenty years after his brief Austrian adventure, he would be invited to live in Paris.
How? it's a long story. On January 25th of 1494, King Ferdinand of Naples died, and King Charles VIII of France contested the excommunicated now-King Alfonso claiming that through his grandmother Mary of Anjou he was the legitimate heir to the Throne of Naples. Ludovico backed Charles' claim, and even housed the King of France and his army in the town of Asti on their way south to depose Alfonso.
However, Ludovico (ever paranoid) quickly took issue with the French rearguard's permanent headquarters in Asti, headed by no less than Louis, the Duke of Orleans who through his grandmother Valentina Visconti had a claim to the Duchy of Milan. Louis probably wasn't going to press the claim, but Ludovico pushed to form Italian League anyway. Consequentially, a Milanese army lay siege to Asti, Alfonso was reinstated in Naples, and a Venetian army chased Charles VIII back to France.
Too bad Charles VIII died childless, and the French throne passed to none other than Louis of Orleans! In 1499 Louis, now King Louis XII, pressed his claim on Milan. Leonardo first fled to Genoa with Ludovico's court, and then became somewhat itinerant around Italy, painting in Mantua, Venice and Florence. Somehow, he managed to come to be employed by Cesare Borgia, the most powerful vassal of the Papacy (and illegitimate son of Pope Alexander VI). In Rome he was employed as a military engineer, and his reputation as a true interdisciplinary genius was established. Apart from examining enemy fortifications, his letters patent (written by Cesare in a mix of Spanish and Italian) entrust him to, "see, measure, and well examine our encampments, fortifications and states [...]" and to do what was needed to strengthen them. Leonardo also invented a new variety of gunpowder at this time.