How did Otto von Bismarck, the chancellor of Prussia, lead the drive for German unity?
With a bigger and better Prussian army, and in 3 short wars (first against Denmark, second against Austria, third against France) he established Prussian leadership among the German States
For a resumé of the 2 first wars of the German Unification, I suggest the following sources
As one of the first acts as Prussia's Chancellor, Bismarck ordered the collection of taxes even without Parliament's consent, to improve and enlarge the Prussian Army.
The economic (rapid growth of Prussian economy), military (advantages of the Prussian Army compared to the Austrian Army) and diplomatic conditions easied his job. As to diplomatic conditions, Britain was interested to remain neuter, France had colonial problems (mostly the Second Franco-Mexican War) and didn't want to intervene earlier and Bismarck had obtained Russia neutrality.
As to the war against the Danes, Second Schleswig War, Prussia was in a better situation than in the First Schleswig War (1848-1851), Bismarck got the help of the Austrian Army and Denmark had just passed by a monarchic succession with a Constitutional crisis in which Denmark tried illegally to suppress Schleswig-Holstein autonomy. Denmark lost the war and the division of spoils between Austria and Prussia led to a casus belli between these former allies.
In the war against Austria, Prussia got the alliance of Italy that wanted Venetia held by Austria. Bismarck assured himself of the neutrality of the other non-German European States Most of the German States sided with Austria mainly of defending their thrones. Because of false reports of mobilization of Italian troops, Austria declared war. Austria lost and a short war avoided intervention of other European states.
As to the Franco-Prussian War, more details in this source
Franco-Prussian War (19 July 1870 – 10 May 1871)
The war against France (a non-German power) was instrumental to rally mostly the Southern German States under the leadership of Prussia.
The immediate cause of the Franco-Prussian War was the candidacy of a Prussian prince to the throne of Spain; even though that was withdrawn under French diplomatic pressure, Bismarck let leak to the press a telegram treating in demeaning fashion the French representative what caused inflamed reaction of the French public opinion. Napoleon III also thought that France had good chances of winning the war.