The first time Vlad III. was connected to the vampire myth was indeed when Bram Stoker decided to center his Dracula novel around him.
During his lifetime, Vlad III. was engulfed in the ever present war between the Western European powers and the Ottoman Empire. The region he ruled as a Voivode (a duke), Wallachia, was a staging ground for armies of both sides and thus much coveted. Vlad indeed spent some time as a hostage at the Ottoman court and switched allegiances repeatedly during his lifetime. During these times he earned both the names "Tepes" meaning "the Impaler" as well as "Dracula" - but the name Tepes only came up after his death and is considered to be slander by his political opponents to brandish him as a brutal ruler. A story about him goes that he once impaled 100,000 people and held a feast beneath their dying corpses. A rather outlandish claim, from the numbers alone. But indeed, impaling was rather common in his days and by no means something that made Vlad stand out in any way amongst his peers.
While it has been theorized that he was named Dracula after the Romanian word "Dracul" or "Dragul", meaning "Devil", it is not contested in scientific circles that it is derived from the Oder of the Dragon to which his father belonged. The Order of the Dragon was a European order of knights who fought the Turks and it's members carried "Dracul" as a cognomen. So Vlad III.'s father was known as "Vlad Dracul" and Vlad III. as "Vlad Draculea" meaning "little Dragon". This most likely led to his cognomen Dracula.
But the region Vlad ruled was not only coveted as a staging ground for armies and a buffer zone between the political hemispheres, but also because the northern parts encompassed a rich area - "Siebenbürgen", an area settled by German settlers and rich in industry and natural ressources. This area was in the Hungarian sphere of interest and led to various conflicts between the Voivodes of Wallachia and the King of Hungary.
Now, during Vlad III.'s time, the King of Hungary was Matthias Corvinus (who would later also become King of Bohemia and lay claim to the Dukedom of Austria). Vlad and Matthias Corvinus switched between alliance and feud, even leading to an imprisonment of Vlad in Hungary, before he was released and managed to reconquer his land another time (the third time to be precise), before being killed in a war against the Turks.
But this feud with the Hungarian king is the root of the demonization of Vlad III. In 1462/63, the first pamphlets began circulating in the Holy Roman Empire describing Vlad as a gruesome ruler and evil person. They were titled "Histori von dem posen Dracol" - the stories of the evil Dracula. They gave colorful descriptions of the supposed deeds of Vlad, for example the aforementioned feast amongst the impaled (but without calling him Tepes yet). But very important here is the origin of these texts: They emerged in Vienna, where at this time the Hungarian king resided to negotiate with the Holy Roman Empire about the Dukedom of Austria. And he brought his hostage at the time with him: Vlad III. of Wallachia. So we have a pretty certain case of political slander at the root of the demonization of Vlad.
Later on, he would be stylized as the "Devil" (in a chronic of the city of Konstanz for example, they would talk about Vlad as the "tüffels sun" - the devil's son). In the arts, he was stylized more and more as a Turk - there is a rather distinct picture of Vlad riding a horse, clad in Turkish armor and carrying dead babies on his lance, wearing the hat and beard typical for a Voivode to distinguish him as Vlad. So political roots of the bad press are rather certain.
Now, when Stoker wrote his Dracula, he searched for a historical figure he could link his main antagonist to, to make him more interesting or graspable.
But, and here comes my own conclusion that I drew when researching Vlad in University, he also wanted to draw from the mystical realm of the "wild east". As I pointed out before, the Wallachian region was heavily contested between Turks and Europeans, leading to it being sort of a "twilight zone" between Orient and Occident. When Stoker wrote Dracula, the Balkan area was once more in the spotlight of international tension: But now between the expanding empire of Russia and the Western European powers (think Crimean War and Berlin Conference).
If we look at the novel Dracula, we see for example that Dracula uses Russian ships to travel to and from England. So Dracula really rather was a supernatural, mystical thing coming from a backwards and mysterious twilight zone between the East and the West to threaten the European people.
And this is very reminiscent of Vlad III., who had to navigate the difficult politics of his time, appease superpowers to the East and the West and was branded a devil, blood drinker and evil person because of it.
Sources (beware, mostly in German):
Arata, Stephen D: "The Occidental Tourist. Dracula and the Anxiety of Reverse Colonization." In: Victorian Studies 33, 1990
Barber, Paul: „Vampires, Burial, and Death. Folklore and Reality.”, Yale University Press, New Haven and London 1988
Engels, Johann Christian: „Geschichte des ungarischen Reiches und seiner Nebenländer“, Volume 4, 1804
Florescu, Radu and McNally, Raymond: „Dracula - Prince of many faces. His life and his times”, Back Bay Books, New York 1990
Goll, Sebastian: „Typisch Balkan! Dracula, Armut und Waisenhäuser.“, in: Gebhard, Gunther (u.A.) (Hrsg.): „Das Prinzip „Osten“. Geschichte und Gegenwart eines symbolischen Raums.“, transcript Verlag, Bielefeld 2010
Harmening, Dieter: „Der Anfang von Dracula. Zur Geschichte von Geschichten.“, Königshausen + Neumann, Würzburg 1983
Haumann, Heiko: „Dracula. Leben und Legende.“, C.H. Beck, München 2011
Haumann, Heiko: „Dracula und die Vampire Osteuropas. Zur Entstehung eines Mythos.“, in: „Zeitschrift für Siebenbürgische Landeskunde“, 2005
Kast, Sheilah and Rosapepe, Jim: „Dracula is dead: how Romanians survived Communism, ended it, and emerged since 1989 as the new Italy.”, Bancroft Press, Baltimore 2009
McNally, Raymond T. and Florescu, Radu: „In search of Dracula: The history of Dracula and vampires.”, Mariner Books, Boston 1994
Miller, Elizabeth: “Vlad Tepes vs.Count Dracula.” In: Miller, Elizabeth: “Reflections on Dracula”, Transylvania Press, White Rock 1997
Ruthner, Clemens: „Süd/Osteuropäer als Vampire: Draculas Karriere vom blutrünstigen Tyrannen zum mythischen Blutsauger.“, in: „Germanistische Mitteilungen“, 2000
Seesslen, Georg and Jung, Fernand: „Horror: Geschichte und Mythologie des Horrorfilms, Grundlagen des populären Films.“, Schüren Verlag, Marburg 2006Seipel, Wilfried: „Dracula. Woiwode und Vampir.“, Ausstellungskatalog, Kunsthistorisches Museum mit MVK und ÖTM 2008
Unterholzner, Bernhard: „Vampire im Habsburgerreich, Schlagzeilen in Preußen. Zum Nutzen des Vampirs für politische Schmähungen.“, 2009: http://www.kakanien.ac.at/beitr/vamp/BUnterholzner1
If you have any further questions, feel free to ask. This is only a very superficial introduction.