I know what Richard did OTL, but Richard is facing a situation where he doesn't have to worry about Philip (or John, if that's a worry) - how much that will change his behavior is something I can't just look up OTL for.
This is information from a book I am reading about the Third Crusade (The book is called The Crusades: The War For The Holy Land by Thomas Asbridge), it is a useful and informative book with much info about all the crusades (I have just finished the part about the Third Crusade) and their leaders. Respond to this thread with specific questions and I will look through the book and find the info you need.
Answering the quote above, since Richard the Lionheart doesn't have to worry about Phillip in league with John attacking the English territories, he would probably stay in Palestine for longer amounts of time, with an attempt to either lay siege to Jerusalem or try and attack Egypt.
Any questions?