In the renaissance, prior to mass public education in Europe, how hard would it be for someone to pull themselves up by their boot straps so to speak? Would a driven but poor child be able to get an education through churches, libraries, ect? How difficult would that be? How rare would it be for someone who was born poor to become an educated, wealthy individual?
Where and who and the political and economic on-goings matter a lot. There's an excellent social history by Emmanuel Le Roy Ladurie that deals with your question very directly.
Ladurie follows a Swiss peasant, Thomas Platter (1499), as he rises through the world by catching on to the new Reformation humanism. Despite beginning his life as a vagabond Platter was able to participate in a fast growing economy of education, typically seeking employment as a teacher no sooner than he had learned material himself. Thomas Platter also benefits from the easy availability of credit-- gold from the monasteries destroyed by the Protestant Reformation or from the New World. By wisely borrowing for investment in an environment of high inflation Platter comes to own several properties. In turn he is able to send his sons to study medicine (a higher form of learning than classics) at the best French universities, and so in two generations education raises the Platters from pigpens to places as court physicians.
Platter was definitely a highly motivated individual, willing to do things like postpone marriage and invest his low wages (working at one point as a apprentice ropemaker) in books and Latin/Greek/Hebrew lessons. His life isn't "easy," but the main thing distinguishing him from his less socially mobile family is his consistent motivation and his luck in choosing a "boom" industry like classical education during the Reformation. In many parts of Europe, definitely central Europe, the Reformation had the character of a broad social revolution, displacing classes and the values that had upheld them, ultimately creating many opportunities for social mobility in the revolutionized Protestant society.
Though Ladurie would never slight Platter's gumption as a historian his work is dedicated to contextualizing Platter's success. Much more could be accomplished in the favorable circumstances of Swiss humanism circa 1530 than say a French village circa 1400, if only because in 1400 far fewer people care about learning Hebrew and none of them would pay good money to a ropemaker who says he can teach it. There's no single answer to your question, but asking it speaks to larger question about social mobility in different European contexts.
There were regulations in medieval London providing for the education of poor students. Many schools were run by churches, but some were independent; there were even "dame schools" run by female teachers. As we can see by this and similar sources, in the late Middle Ages it was not too difficult for anyone in a town to get a basic education. This applied largely to commoners who intended to become merchants, lawyers, etc., but also to ordinary peasants who just wanted to be able to read their business contracts: there was a very active land market in the late Middle Ages.
It is a common myth that only people training to be clergy were educated; in fact, many children--both boys and girls--got a good basic education. This is testified to by the many surviving letters and wills written by both men and women. Universities, of course, then as now, were harder to get into. Also they, unlike the preparatory/grammar schools, did not accept women.
P.S. this is really a medieval answer, not a Renaissance answer, but your time period includes two centuries of the Middle Ages so here it is