Not very many. I'd have to do some serious digging to come up with actual proportions, but in general Richard's crusade drew almost exclusively from the upper classes of English society. The Third Crusade was the first one to see a majority of crusaders take the sea route to the Holy Land as both Richard I and Philip II set off from Southern France by ship. Sailing across the Mediterranean was a faster way to reach the Holy Land, but it was also significantly more expensive. Due to the cost of paying for a berth on the ship (in most cases nobles would pay for the berths of the soldiers in their retinues, although in some cases they might have to receive gifts from the king to subsidize their expenses) peasants were largely unable to go on the Crusade. That is not to say that none did, the fervor of crusade inevitably encouraged some people to go, but by and large the Third Crusade lacked vast peasant/non-soldier contingents, at least in the French and English armies.

That said, not everyone who went on Crusade was a noble. As mentioned above nobles would pay for their soldiers to go on crusade and many of these individuals would have been of lower rank. In general, though, these would have been the sort of men who had some experience or skill as soldiers (not usually farmers, for example). In some cases they may even have been mercenaries, essentially hired for the crusade, although officially going out of 'religious enthusiasm' (and inevitably in part motivated by religion as all medieval people were).

Christopher Tyerman's How To Plan a Crusade features a lot of interesting details about the logistics and practicalities of details, but the breadth of what it covers means he doesn't devote a lot of time to the specifics of Richard's crusading army.