I think you have a bit of a misconception going on here. Martin Van Buren wasn't a poor immigrant who battled xenophobia - he was the son of native-born Abraham Martensen van Buren (a captain in the Albany County Militia during the Revolution), who was the son of native-born Marten Pieterse van Buren, and you have to go back to Cornelis Maessen van Buren, born in the early seventeenth century, MVB's great-great-grandfather, to get to a Van Buren who came over from the Old World. On both sides of his genealogy, there were deep roots in New Netherlands and therefore America. He spoke Dutch as a first language because the Dutch descendants of Albany-area towns like Kinderhook and Valatie (and even of cities like Albany and New York, to some extent) chose to keep their customs and not assimilate into English society.

In the late seventeenth century, after New Netherlands was taken over by the English, the Dutch inhabitants were seen as a group that needed to be appeased, not subjugated. Albany, the former Fort Orange and the northern outpost of the colony, very much remained a Dutch city, filled with prosperous and independent Dutch people; there were concerns that they might rebel and allow the French through the border. Eventually, more English people migrated to New Amsterdam/New York City - disproportionately at the higher levels of income. And here's where it gets very telling. Around 1700, Dutch men were over whelmingly, like 99.5%, likely to marry a Dutch woman, but English men, while not entirely exogamous, were more likely to marry a Dutch woman than a English woman. So on the one hand, you have a Dutch community that's determined to be insular, and you have a well-to-do English community with substantial kinship ties to the Dutch one. The colony/state would continue to become more and more English (and other ethnicities) and less and less Dutch-dominated, but in the upper class, English and Dutch were still closely intertwined, and the Dutch families were very much dominant. General Philip Schuyler, for instance, was not only a national name but a New Yorker of purely Dutch ancestry; the Van Rensselaers, the Van Cortlandts (the general's mother was a Van Cortlandt, btw), the Ten Broeks, the Beekmans, the Dyckmans, and more were among the biggest names that still resonate today. Even "English" families like the Livingstons were still quite intermarried with the Dutch. This isn't just a case of a few powerful "holdovers", but the overall landscape of New York politics. Even if you weren't from one of these well-known families, being descended from New Netherlands Dutch stock would read extremely well in New York. The first assumption would not be "oh, some immigrant."

As for the answer to how he managed to become president, Van Buren started out in New York politics, where all of this background worked in his favor. He served as the state attorney general, and then was elected to the Senate to represent New York - and then as now, Congress is a way to use a local-politics campaign to achieve national prominence. Getting to the Senate required the New York-specific connections, but it allowed Van Buren to make connections with politicians outside of the state, to make speeches that would be reported around the country, and serve on committees that had nothing to do with New York. As I noted in an earlier answer about his relationship to slavery and abolition, he also practiced politics with an eye for compromise and, well, politics, rather than for a greater goal. Massively involved in party politics, Van Buren did the work to assist others, like Andrew Jackson, in tacit exchange for reciprocal support later. After being elected governor of New York, he was quickly appointed Secretary of State under Jackson, and he was Jackson's running mate in his reelection campaign, which made him vice president. From there, it was a relatively easy stepping-stone to the presidency.