The qualifications of the Messiah are to be a male patrilineal descendant from the House of David (and of course to have a Jewish mother). To be very observant of the Written and Oral Law. To bring all the Jews back to following the Torah. To build the Temple. Gather the dispersed Jews back to Israel. And to cause the world to turn to the Jewish G-d.
Of course, there's no way to know his genealogy today, so for that we'll have to rely on his having fulfilled all the other things. And it does automatically exclude any Priests or Levites.
Rabbi Schneerson was the Grand Rabbi of Chabad/Lubavitch. Chabad/Lubavitch is one of the top two largest Hassidic groups in the world. During his lifetime, the Rabbi made great inroads in fulfilling some of the requirements the Messiah would fulfill, leading his followers to believe he was the Messiah. After his death, there remains a sub-group of that sect that still believes he will be resurrected to be the Messiah and another group who believes he hasn't even died.
The comparison of this belief to Christianity has been made many times. Of course, a key difference is that he actually worked to bring people closer to G-d and to Jewish Law, as opposed to closer to worshiping a man. Still, the nonsense about being resurrected is still close enough to Christianity to make everyone else uncomfortable around them (and bear in mind that even though they're the largest Hassidic sect, they're still a fraction of the entire Orthodox and even Hassidic world). And although there's not numbers on it, it's also not the entire sect who believes these things, just a large portion of them.
The Grand Rabbi of Chabad was very observant of the Written and Oral Law. He worked very hard to bring Jewish back to following Torah. He also put effort in brining non-Jews to turn to G-d. There had not been anyone as successful as he was in a very long time. So there was definitely reason to think he potentially the Messiah at the time. However, like Bar Kochba, partial fulfillment isn't enough and so he was disqualified with his death.
