Jean de Joinville was not your typical medieval chronicler. He was not a monk or a scribe, instead he was a knight who was a part of King Louis IX of France. He followed the king on the Seventh Crusade and was involved in the battles of that failed campaign. While he does not go into great detail of the battles he does recount them with more flavour than is typical.

Geoffrey of Villehardouin is another knight historian. He was involved in the Fourth Crusade and recounts the battle of Constantinople, in which he was militarily involved.

It is a bit later than you are looking for, but Bernal Diaz's account of the capture of Tenochtitlan in Conquest of New Spain is pretty terrifying.

Hmm, the memoirs of Usamah Ibn-Munqidh are quite an interesting read on Crusader warfare; it's not so much a history of specific battles but a recounting of his experiences in a variety of battles(pitched line battle, skirmishes, ambushes, palace coups) etc. and discusses his ideas about what makes someone effective in battle(and interestingly enough, copious discussion of battlefield medicine and surgery). They've been published in English as "Memoirs of A Syrian-Arab Gentleman".

The Chronicles of Albertus von Aken (Albert of Aix-la-Chapelle or Albert of Aachen in English) about the first crusade are an interesting read, I made a study of them. He wasn't a first hand observer but he talked to many people going and returning from the crusades and told of the hardships of the trip and the live.

There are several accounts of the siege of Constantinople in 1453, many of which were firsthand.

Look to George Sphrantzes' "Chronicon Minus." The "Chronicon Maius" is also interesting, but it wasn't written by Sphrantzes. See also Archbishop Leonard of Chios's account of the siege, which was written within months of the fall of the city. Nicolo Barbaro was present during the siege, and his chronicle of the battle may be the most accurate we have. Giacomo Tetaldi and Ubertino Pusculus were other Italians present during the battle who wrote useful accounts.

Laonikos Chalkokondyles wrote an account of the siege (and more) roughly contemporaneous with the events, but the text isn't very good. Kritoboulos wrote a useful account that wasn't discovered until the 19th century. Doukas also wrote a useful history.

These accounts generally aren't going to be a war-fiction style account of a battle, but they certainly are an account of a horrible battle with mass casualties. Barbaro wrote about how, after the city fell, the Golden Horn was so filled with heads it reminded him of melons bobbing in the Venetian canals. Most of the battle scenes involve siege warfare as well as hand-to-hand combat in the peribolos, which was the area between the inner and outer city walls.

This is actually much older than the European Middle Ages, and isn't really a battle, but a training exercise (in which lives were lost). It definitely captures a certain atmosphere.

Sima Qian (who died around 80 BCE) included in his grand history of China the following biography of Sunzi, the general who wrote the book known today as The Art of War. I'm pulling this from wengu.tartarie.com's introduction to Sunzi:

Sun Tzu Wu was a native of the Ch`i State. His ART OF WAR brought him to the notice of Ho Lu, King of Wu. Ho Lu said to him: "I have carefully perused your 13 chapters. May I submit your theory of managing soldiers to a slight test?"

Sun Tzu replied: "You may."

Ho Lu asked: "May the test be applied to women?"

The answer was again in the affirmative, so arrangements were made to bring 180 ladies out of the Palace. Sun Tzu divided them into two companies, and placed one of the King's favorite concubines at the head of each. He then bade them all take spears in their hands, and addressed them thus: "I presume you know the difference between front and back, right hand and left hand?"

The girls replied: Yes.

Sun Tzu went on: "When I say "Eyes front," you must look straight ahead. When I say "Left turn," you must face towards your left hand. When I say "Right turn," you must face towards your right hand. When I say "About turn," you must face right round towards your back."

Again the girls assented. The words of command having been thus explained, he set up the halberds and battle-axes in order to begin the drill. Then, to the sound of drums, he gave the order "Right turn." But the girls only burst out laughing. Sun Tzu said: "If words of command are not clear and distinct, if orders are not thoroughly understood, then the general is to blame."

So he started drilling them again, and this time gave the order "Left turn," whereupon the girls once more burst into fits of laughter. Sun Tzu: "If words of command are not clear and distinct, if orders are not thoroughly understood, the general is to blame. But if his orders ARE clear, and the soldiers nevertheless disobey, then it is the fault of their officers."

So saying, he ordered the leaders of the two companies to be beheaded. Now the king of Wu was watching the scene from the top of a raised pavilion; and when he saw that his favorite concubines were about to be executed, he was greatly alarmed and hurriedly sent down the following message: "We are now quite satisfied as to our general's ability to handle troops. If We are bereft of these two concubines, our meat and drink will lose their savor. It is our wish that they shall not be beheaded."

Sun Tzu replied: "Having once received His Majesty's commission to be the general of his forces, there are certain commands of His Majesty which, acting in that capacity, I am unable to accept."

Accordingly, he had the two leaders beheaded, and straightway installed the pair next in order as leaders in their place. When this had been done, the drum was sounded for the drill once more; and the girls went through all the evolutions, turning to the right or to the left, marching ahead or wheeling back, kneeling or standing, with perfect accuracy and precision, not venturing to utter a sound. Then Sun Tzu sent a messenger to the King saying: "Your soldiers, Sire, are now properly drilled and disciplined, and ready for your majesty's inspection. They can be put to any use that their sovereign may desire; bid them go through fire and water, and they will not disobey."

But the King replied: "Let our general cease drilling and return to camp. As for us, We have no wish to come down and inspect the troops."

Thereupon Sun Tzu said: "The King is only fond of words, and cannot translate them into deeds."

After that, Ho Lu saw that Sun Tzu was one who knew how to handle an army, and finally appointed him general. In the west, he defeated the Chu State and forced his way into Ying, the capital; to the north he put fear into the States of Chi and Chin, and spread his fame abroad amongst the feudal princes. And Sun Tzu shared in the might of the King.

If you're looking for something online, the website Eyewitness to History might have what you're looking for. They have a section for the Middle Ages and Renaissance, which includes a few battles during the Crusades and others from the point of view of knights and soldiers etc. I hope that helps!