WARNING: On some parts of this quest if you kill certain mobs you will tend to set back the favor you are trying hard to earn. For example, when you are trying to prove your worth to the crown if you kill Saladin or King Richard it costs you favor. So it is best to finish each part before killing other mobs.
Starting the quest:
Find Philip, ask philip golden fleece (Or order), ask him yes
There are three parts to this quest.
1st part involves killing Saxon Outlaws around Stag area (Other mobs are included in said killing will note them down as it would take a few rounds of repops to get enough to progress)
Once it says you have proven yourself, go back to Philip and he will congratulate. Ask him task to start next round.
2nd part you can get by killing most of the pirates around Wu Chen, including Chen himself. Most of the time doing this I have been able to get all that was needed in one round. (Other Mobs possible will note as I recall)
Once you have proven yourself, go back to Philip and he will congratulate you. Ask him task to start the final round.
3rd part takes a bit longer, I only found Ghents at the entrance to Flanders raises this one. Will do more research to find other mobs as this one tends to take the longest in my experience.
Additional mobs for Chivalry: The vampire in Port of London, Captain Renault and Major Strasser in Casablanca, the man-eating lion in Savanna.
Finish: Go back and he will congratulate you. He will then ask you what you wish to have. Ask him Collar, Emblem, or Badge.
You can also do quests to raise favor as well.
Alan-a-dale Love quest and Aladdin Love quest both raise favor for Chivalry.