No, the immersion is just a dunk in the water. It doesn't require any kind of scrubbing. The Rambam writes in Hilchot Mikvaot 1:2 כָּל מָקוֹם שֶׁנֶּאֱמַר בַּתּוֹרָה רְחִיצַת בָּשָׂר וְכִבּוּס בְּגָדִים מִן הַטֻּמְאָה אֵינוֹ אֶלָּא טְבִילַת כָּל הַגּוּף בְּמִקְוֶה. "Anywhere the Torah speaks of washing the flesh or washing clothes to remove ritual impurity (tumah) it refers to immersing the whole body in a mikvah."
In terms of immersing with clothes on, the immersion should ideally be done without any clothes on. There is a principle called chatzizah which means a physical barrier between the skin and the water. The Talmud in Sukkah 6a-b learns this from the verse in Leviticus 14:9 “And he shall bathe his flesh in the water”--meaning the flesh comes in direct contact with the water. In some cases immersing with clothing can work, see Beitza 18a, but it's a complicated topic in practice. It's going to depend on how loose the clothing is, how well the water penetrates, etc. Google turns up a shiur (recorded lecture) on the topic in practice, which I have not listened to. https://etzion.org.il/en/talmud/seder-moed/massekhet-beitza/beitza-18a-tevila-clothes.
Objects other than people (including clothing) sometimes needed to be immersed to remove ritual impurity. In a case where both a person and his clothing needed to be immersed, the person and clothing would have been immersed separately. By the way, most of the cases of immersion you will read about in the Bible are not practiced nowadays because without the Red Heifer we are all in a state of incurable tumah impurity anyway. In addition, most people only needed to be ritually pure (tahor) when working in the Temple, bringing a sacrifice, or making a Temple pilgrimage, see Rosh Hashanah 16b, none of which apply nowadays. The only biblical applications of immersion nowadays are (1) a woman purifying from a state of niddah; (2) conversion; (3) ascending the Temple mount. Some people also immerse for purposes of sanctification or purification that are not directly related to biblical issues of tumah and taharah (usually men and most often before sabbath and holidays). But none of these immersions involve immersing the clothing and, therefore, are usually done without clothing.