Considering he was not a major nobleman, and considering the dearth of historical sources surrounding really ANY medieval figure, we actually know a relatively large amount, if the sources can be trusted. The advantage with Wallace is that he is attested in several different accounts of the period, which makes it unlikely that he didn't exist at all.

There is not necessarily a general consensus about where he was from or who is parents were, but given that he was probably not from a high ranking noble family this is unsurprising. What is not disputed is that he likely came from a landholding family somewhere in the vicinity of Lanark in Scotland. Very (VERY) little is known about his early life. The first solid mention we have of him is when he murdered the sheriff of Lanark, essentially a military-political leader in the area. However, at the time England had asserted dominance in the region (under the guise of bringing stability after the death of Alexander III and the instability of John Balliol's rule) and so the attack was wrapped up in the ongoing conflict between the English and Scots.

I won't go into the historical background of the Scottish Wars of Independence, but essentially William Wallace took advantage of the general inability of the English to project power into the peripheral areas of Scotland, as well as Andrew Moray's simultaneous uprising in the North to begin fighting the English. Wallace is seen as a freedom fighter in Scottish sources and a brigand by the English writings, so it's hard to tell which he really was. The answer is probably somewhere in the middle. Living off the land and gathering a force of Scots (mostly lower-class, as the nobles had officially sworn allegiance to Edward I) he proceeded to disrupt the English and Pro-English Scots.

Despite many more nobles and military leaders submitting to the English, Wallace and Moray refused to surrender and instead joined forces, besieging Dunbar. This is all relatively well attested. The battle of Stirling Bridge is also comparatively well recorded, especially considering that many medieval battles are very vaguely described in the chronicles. It seems that Wallace and Moray waited until about half the English army crossed the bridge and then descended from the highground to attack the vanguard, destroying it before the remaining troops could cross the river. It was a stunning victory in every aspect, though some credit must go to the incompetency of Hugh de Cressingham. It should also be noted that a large part of English strength was tied up in a cotstly war with France at the time, and Edward I (a capable military leader who I would love to describe in better detail) was unavailable to lead the troops. In any case the battle meant that English presence in Scotland was now limited to a string of fortresses, and the countryside was now in the hands of the victorious Scots.

At this point it becomes clear that Wallace had gained a significant reputation, and with the death of Moray (from wounds suffered at Stirling) he was one of the sole capable military leaders in Scotland. In addition, he still possessed the greater part of the troops he had gathered during his rebellion, which lent him considerable political weight.

Some scholars have pointed out that Keeping his army together was probably a significant reason why Wallace proceeded to raid northern England. While this may seem unprecedented, it was a viable alternative to disbanding his army or foraging in Scotland - instead he took supplies from the now undefended English, and could sustain his army at their cost. In addition, putting pressure on the Northern shires had previously been a primary strategy of Scottish kings, and would continue to be a prominent tactic in later years. He did not besiege York, but he probably caused a significant scare to the English, who put aside their differences (there had been some disturbances with the Earls of Hereford and Norfolk, but that's for another day) to fight him.

Edward was quick to organize a new expedition after he returned from the continent, and gathered what was probably the largest English army up to that point. Records show that at least 20,000 infantry and 1,000 cavalry were present days before the battle, and this is an unprecedented force. Comparatively Wallace could probably field only around 5-6,000. You can understand the disparity in numbers.

Wallace was hesitant to fight the English in a pitched battle, and preferred to wait until English supplies ran out before pursuing them back to England. This was almost successful, but in the end Edward was able to strike at Wallace's army at the battle of Falkirk.

I won't got into the details of the battle, but it ended with a crushing victory for the English. Most of Wallace's army was destroyed, and while he escaped he no longer possessed the political clout it had provided. In addition, many Scottish nobles once again did homage to Edward, undoing much of the work of his rebellion. In the end he was arrested and executed on Edward's demand, for treason against the English people - basically a trumped up charge, but as Edward probably considered the Scots to be under his rule it makes sense how he could spin any rebellion to be a treasonous act.

That's most of what we know. It comes from a variety of sources, so I'll provide most of what I know. The Lanercost Chronicle is a primary source which has accounts of much of the struggle between England and Scotland. You should be able to find translations online, but a physical copy is usually restricted to large libraries. The Chronicle of Walter of Guisborough also relates some of the history surrounding the period, and much of our information concerning Wallace's movements into England are found here. For a look at official records, I would highly recommend The Welsh Wars of Edward I by Morris - it focuses on the Welsh Wars but still goes over much of the daily movements and logistics of Edward's Falkirk campaign, plus it's probably one of the best analysis of the payroll accounts of Edward I's reign out there. Some of our information comes from John of Fordun, and you can find some of his works online. Unfortunately Fordun was writing around 50-60 years after the events themselves, but is still a valuable source.

Hope some of this was helpful, let me know if you have any more questions. Edward I and the events surrounding his reign are my specialty so ask away!