Before I get into answering this question I feel like I should talk about the word "feudal" for a second. Simply put, there was no united "Feudal System" throughout the Middle Ages either across nations or across time, the idea that is traditionally associated with the word feudalism simply didn't exist, so when we talk about things or ideas within the medieval systems of government we should always be extremely specific about a time and place, perhaps even under a specific ruler, as there were some major changes to systems of governance between monarchs (See: Henry III and Henry IV of Germany).

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With that in mind I'll answer your question within the context of the Ottonian Dynasty, specifically from 962-1024, as it could be held up as emblematic of the period, I also happen to be well read on the subject.

So. when you ask if a serf could become a knight I'll assume you're asking if a serf could rise to nobility. In this time period, probably not. The idea of nobility in the Early Middle Ages was extremely stratified. I personally can't find any good evidence of such an event happening during this period; however, the Ottonians did perpetuate a fascinating institution from the Carolingian Era, that of the missi dominici, or messengers of the lord/ruler, which were essentially serfs who were also knights. They did not have noble status, but they were literate and fairly wealthy. This practice, and these serf-knights, seems to be quite unique to the Carolingians and the Ottonians, so while a serf may not have been able to obtain nobility he could become wealthy and literate and perform vital functions of state that were normally reserved for nobility, and so could functionally become a noble during his life-time, but this prestige and ability could not become hereditary.

Sources;

Warner, David A. "Ottonian Germany: The Chronicon of Thietmar of Merseburg." Manchester University Press, 2013.

De Corvey, Widukind. The Deeds of the Saxons. Translated by David Bachrach and Bernard Bachrach. Catholic University of America Press, 2014.

Fichtenau, Heinrich. The Carolingian Empire. University of Toronto Press, 1978.