Are there any records written by Christians about Muhammad that were written during his life? If so, what did they say about him? I would be very surprised if the Church of ~600-630 AD didn't have something to say about a man claiming to be the prophet of God.
The most notable early Christian writing that I am aware of was by John of Damascus (676-749). In Heresies in Epitome: How They Began and Whence They Drew Their Origin he claims that the basis of Mohammed's fundamental teachings came from an Arian monk:
There is also the superstition of the Ishmaelites which to this day prevails and keeps people in error, being a forerunner of the Antichrist. They are descended from Ishmael, [who] was born to Abraham of Agar, and for this reason they are called both Agarenes and Ishmaelites. They are also called Saracens, which is derived from Sarras kenoi, or destitute of Sara, because of what Agar said to the angel: ‘Sara hath sent me away destitute.’ These used to be idolaters and worshiped the morning star and Aphrodite, whom in their own language they called Khabár, which means great. And so down to the time of Heraclius they were very great idolaters. From that time to the present a false prophet named Mohammed has appeared in their midst. This man, after having chanced upon the Old and New Testaments and likewise, it seems, having conversed with an Arian monk, devised his own heresy. Then, having insinuated himself into the good graces of the people by a show of seeming piety, he gave out that a certain book had been sent down to him from heaven. He had set down some ridiculous compositions in this book of his and he gave it to them as an object of veneration.
The rest of the text can be found here.
Some say that the iconoclast heresy in Byzantium arose because certain Christians were envious of the material success of surrounding Moslems and surmised that their practices - especially their rejection of depictions of human form - were earning them God's favor. John of Damascus was one of the most vocal critics of the iconoclast Emperor Leo the Isaurian. Ironically, John wrote his Apologetic Treatises against those Decrying the Holy Images while under the protection of a nearby moslem caliph.