First to provide a little historic background, combining some info I wrote from earlier comments.

Weimar institutions were unraveling under political and economic strain in the early 1930s. Right-wing authoritarianism was definitely on the rise in popularity, but that did not inevitably mean Hitler and the NSDAP inevitably would gain power and establish the dictatorship it did (let alone start a world war and the Holocaust). Henry Ashby Turner's Hitler's Thirty Days to Power gets into the contingencies of the actual seizure of power in January 1933. The NSDAP was the largest party in the Reichstag, but was not able to form a government on its own. Hitler refused to participate in any coalition government unless he personally was Chancellor. Two elections in 1932 had seen political deadlock, and the second election saw the NSDAP lose millions of votes and face financial straights. A few SA leaders were breaking off from the party to form their own independent organizations. Hitler was actually talking to his subordinates of suicide.

In the event, he ultimately gained the Chancellorship in January through complicated negotiations whereby the sitting Chancellor General Kurt von Schleicher was betrayed by Franz von Papen, who urged President Paul von Hindenburg to overcome his dislike for Hitler and offer him the Chancellorship, with von Papen as Vice Chancellor. The cabinet only had three NSDAP members initially: Hitler, Wilhelm Frick as Minister of the Interior, and Hermann Goering as Minister without Portfolio. The other cabinet ministers were non-party aristocrats largely holding their jobs from earlier cabinets, plus members of the far-right German National People's Party, which was a coalition partner. The cabinet appointment was largely shrugged off by the German public, and von Papen told von Hindenburg that "in two months' time we will have squeezed Hitler into a corner until he squeaks". That things worked out differently were the results of further contingencies of history.

Regarding NSDAP election wins:

The NSDAP was pretty marginal throughout the 1920s, with Hitler and much of the senior leadership imprisoned and the party temporarily banned after the 1923 Putsch attempt. Even after the ban was lifted and Hitler had set about reorganizing the party, it was a fringe group, getting just 2.8% of the vote and 12 Reichstag deputies in the 1928 elections.

By 1930 the situation was vastly different, and the party increased its vote share by eight times: 18.25% of the vote and 107 deputies, to become the second largest parliamentary bloc after the SPD (the Communists were the third largest with 13.13% of the vote and 77 deputies). In the July 1932 elections this support grew further to 37.27% of the vote and 230 deputies, making the NSDAP the largest group in parliament. The November 1932 elections saw a fall in support to 33.09% of the vote and 196 deputies, but it was still the largest party in the Reichstag.

It's worth noting that the NSDAP did have something of a ceiling on its support. Even in the decidedly unfree but still multiparty elections of March 1933, with Hitler as chancellor and massive SA violence and intimidation at the polls (plus the jailing of much of the Communist party), the NSDAP still only received 43.91% of the vote and 288 deputies. It still needed the far right DNVP to govern with a majority, and needed the support of the Catholic Center Party to get the necessary supermajority to pass the Enabling Act.

OK, so that's some background. But specifically what was the role of the Chancellor?

A Chancellor in the German context (both in the Weimar system and in the postwar federal republic) is analogous to a Prime Minister in a Westminster Parliamentary form of government. There really is not an equivalent in the American Presidential system: in the Weimar case the Chancellor was the head of government (presiding over the Cabinet that ran the government), while the Reichspräsident was the head of state. The president appointed the chancellor, but the chancellor needed a majority vote of confidence among the Reichstag in order to serve. An English translation of the Weimar constitution is available here, and the relevant sections describing the Chancellorship are Articles 52-58, where it describes how the Chancellor heads the national ministry (ie the Cabinet), determines policy, and presides over Cabinet votes. Even in areas where the President had prerogatives beyond those of a purely symbolic head of state (such as representing the country in foreign affairs, and serving and commander-in-chief, all such orders had to be countersigned by the Chancellor or relevant minister.

It's worth noting the one gigantic loophole in the Constitution, the infamous Article 48, where the President "may take the necessary measures to restore public safety and order; if necessary, with the aid of armed force." As the Reichstag increasingly became deadlocked after 1930, there were increasing calls (especially by conservatives like von Papen) to institute "presidential rule" where the President assumed control of the government through an emergency decree. Technically, according to the constitution, the Reichstag could override Article 48, and so even this ultimately needed some sort of Reichstag support (hence reaching out to the NSDAP).

A final note is Article 51, whereby in the event of death or disability of the President, the Chancellor assumes the acting position of President. This is essentially what Hitler did when Hindenburg became seriously ill in 1934 and died on August 2 of that year.