To receive the quest, the PC must fulfill two conditions: First, (s)he must be non-chaotic; second, (s)he must have killed 500 monsters. If the PC talks to the old barbarian while these requirements are met, (s)he will be instructed to kill twenty of the kind of monsters that was his/her first kill.
If the player has taken note of their first kill, this is as easy as finding twenty more and killing them, then returning (though the old barbarian will only talk to non-chaotics, so chaotic players that completed the quest objective will still have to convert to get the reward). If the PC's first kill is a monster that only appears once or twice in the game, like a half-orc bouncer or raider lord, the Courage quest is impossible to complete excepting by wishing for the monster about 20 times; since Courage itself can be wished for (and is not terribly crucial) that would be a lousy way to spend wishes.
An easy way of taking note of your first kill is to rename the PC to the monster's name until the Courage Quest is complete. Another method is to search for the first kill in your monster memory. Since the last monster you searched for in the monster memory persists even after saving, you will always be able to recall it from there. The downside is that it makes using the monster memory for its intended purpose slightly inconvenient since you've got to search for the first kill every time after you use the monster memory for other purposes, or else risk losing track of it.
Researching the first kill Edit
If the player has not taken note of their first kill, the old barbarian provides a way of researching it. A list of possible first kills should be made; i.e. the player should try to remember where they started the game and which monsters the first few could have been; this list will usually include goblin, kobold, rat, giant rat, skeleton, zombie, hobgoblin and so on. Then these should be systematically checked off; the PC will venture to the nearest early game dungeon - usually D:1 - and kill one listed monster, but refrain from killing any of the others on the list. Then (s)he can immediately return to the old barbarian. If he reacts saying "Ye art showing promise but ye need to show more" you have the correct monster, as this message signifies that the player has at least made some progress toward the quest, i.e. killed at least one monster. If he doesn't react this way, but makes a critical remark and eyes you suspiciously, you can check off that one monster type and return to the dungeon.
If the player is pretty sure that the first kill was a very common monster, (s)he can also try simply continuing to play as normal and talking to the old barbarian once more. Finding out the first kill using him is more critical if the player has not noted their first kill, but is planning an ultra ending, for which some knowledge of Filk's approximate location is valuable, if only to make sure that he is not already located impossibly deep and the ultra plans should be abandoned.
Alternate reward Edit
If the player has started and finished the quest, but gained the Courage skill in some other way (either a wish, a scroll, or a potion of education), the old barbarian will react surprised and grant a regular two-handed sword as the reward instead.