If Jews converted to Christianity in Nazi Germany then were they still persecuted? How exactly did the government determine who was Jewish or not?

In the early days of the Nazi movement, goon squads like SA and SS went about their way by local knowledge to terrorize Jewish populations. That effectively meant they were basing someone's Jewishness on guesswork and on local informants who could then deceive them to either protect Jews or to screw over non-Jews. SA and SS units could also use their newlyfound power to settle their personal scores by denouncing non-Jews as Jews and then messing them up.

But when the Nazis came to power and when ethnic conflict became legalized repression, the leadership needed a clear definition of who was a Jew and who wasn't. This definition came to its final and most infamous form with the publishing of RGB I #100 and its related pieces of sub-legislation, better known as the Nuremberg Laws.

Your grade of Jewishness was to be determined by the Jewishness of your immediate ancestors, that is your parents, your grandparents and your great-grandparents.

If all eight of your great-grandparents were non-Jews, you were considered a Deutschblütiger, "German-blooded". You also had a slight edge of absolute Aryan purity over anyone who was 1/8 Jewish, i.e. had a Jewish great-grandparent whose child (the subject's grandparent) was not a practicing Jew themselves. A 1/8 Jew would still be considered a full-on Deutschblütiger, but of course they lost their bragging rights of being racially pure. Bummer.

Anyway, the middle category below Deutschblütiger then was Mischling, which is a term from dog-breeding and means "hybrid". Another translation sometimes seen is "half-blood" or "mixed race".

If you were 1/4 Jewish - with a Jewish grandparent - you were mixed race of the second degree. If you were 3/8 or even 1/2 Jewish, you were mixed race of the first degree.

The mixed race individuals were still eligible to hold German citizenship, but first degree mixed races could be "demoted" to being a Jew if they were actively practicing the religion or if they married a Jew.

Which brings us to the lowest classification. 5/8 or more Jewish made you a full-on Jude, Jew. You were no longer eligible to citizenship and you had ever-increasing legislative discrimination cast against you. You were not allowed to serve as a judge, you were not allowed to hire non-Jewish women under the age of 45, you were not allowed to fly the German flag, you were not allowed to marry a Deutschblütiger.

So, to sum it up:

If Jews converted to Christianity in Nazi Germany then were they still persecuted?

Yes. Nazi persecution of Jews was not some abstract religious criticism. It was bigoted, unhumane and despicable hatred. Being a Jew was a sin there was no recovery from.