Depending on where he lived and in what time period, he could accomplish something better but most likely not in his own lifetime.

In Charlemagne's times a freeman (meaning someone that owned the land he worked, no matter how small) could virtually start buying off land from someone who was in deep debt and stack it up to create a property of modest size. In this period (8-9th centuries) he might get some recognition from the local royal officer (the future counts) or abbey as a person of note.
Something along these lines would happen much later, during the 1300s and 1400s with wealthy farmers actually holding reasonable power in the rural zones and being treated better than their peers - although never as an equal of the aristocracy.

In the cities of Northen Italy which were production and proto-industrial centres, farmers from the villages around the city could move in and start working as textile labourers. Hypothetically, after a couple of generations, his son or grandson could learn one of the trades within the urban corporations and enter the political system (which were randomly sorted among the citiziens with a set amount of wealth).
Similarly, his descendants could attend one of the public schools within these cities (paying a "tuition fee") and actually learn the basics of writing and calculus and then become a merchant or notary apprentice - although this last instance is highly unlikely.

As far as war is concerned, usually poor people would arm themselves as best they could and fight as a spearman, especially if they lived in a city. To accomplish something noteworthy he would have needed the ransom of a captured noble, which happened. A change in this situation would occur between the late 14th century and during the 15th and 16th, with the apex of the mercenary companies during the Renaissance and Early Modern Age.

A last option but the most plausible one would have been to join the clergy and work their way up, either by becoming a priest and possibly getting as high as bishop or as a friar and move up within the monastic orders. People as Salimbene de Adam and Jean de Joinville write in their memoirs of having met bishops and confessors of aristocrats which were born in poverty.

In the end, a very shrewd peasant could reasonably become a clergyman and easily go up or could move into a city and lay down the work for his children to raise in status.

Sources:
Vallerani, M., Provero, L., "Storia Medievale", Mondadori Education, 2016;
Mallett, M. " Mercenaries and Their Masters: Warfare in Renaissance Italy", Pen & Sword, 2009;
Petrucci, A. "Breve storia della scrittura latina", Bagatto Libri, 1992;
Le Goff, J. " L'Homme médiéval", Seuil,1988

You speak of a peasant being able to climb the social ladder of the Middle Ages and rise the rungs to the highest position in the land, that of King. I'll do you one better though. How about a peasant (though we need to remember that peasant =/= poor) rising to become Basileus Rhomaioi, Emperor of the Romans, and the most powerful ruler in the world at the time of his apex of power?

Basil the Macedonian came from humble, though not necessarily impoverished, peasant origins in the Theme of Macedon in Thrace (the area between Istanbul and Northern Greece today) from uncertain ethnic origins. Anthony Kaldellis has made a lot of noise recently about the lack of ethnic variation in the Empire, and the primacy of the idea of Roman identity (meaning Greek speaking Christians who followed the Patriarch of Constantinople) overriding all others. Despite this, his origins are sometimes given as Armenian or Slavic and its uncertain what his exact background was.

His early life is somewhat unclear but he entered into the service of a prominent noble and made a name for himself through military action as well as adept social maneuvering and entering into the good grace of both the Imperial family and other prominent landowners. He was able to gain the favor of vastly more wealthy imperial figures who would patronize, and help fund, his rise to prominence. This culminated in his appointment as a member of the personal retinue of the emperor. However he was not done yet! He was even able to rise to the level of marrying the emperor's, Michael III, favorite mistress (though the son she had ostensibly with Basil, the future emperor Leo IV is, and was, widely believed to be the son of Michael III). Even this was not enough for his ambition and he turned on his own patrons rather deviously.

He had the emperor's uncle assassinated based on the fear of his own rebellion for the crown and was made co-emperor after the deed was done (perhaps to ensure that Michael's biological son would ascend to the throne), not satisfied with even this highest possible honor, he had Michael III murdered scarecely a year alter and became the sole ruler of the Byzantine Empire. Not bad for a peasant from rural Thrace eh?

However, as grand and rapid as Basil's ascent was we need to remember that his rise was enabled by people far more powerful than he was originally. His rise through the ranks was predicated on noble largess that he was able to tap into through his own considerable skill at arms and adept social maneuvering, but he did not grow from rags to riches purely on the basis of his own merit. However his story does show that social system of the Byzantine empire was not nearly as rigid as commonly imagined and that it was possible, if extremely rare for peasants to rise to the most powerful positions in the Medieval world.