Howard is the halfling and a level 10 thief who fights kobolds with a dagger.
Leonard is the hobbit and a level 16 merchant who fights orcs with a crossbow.
Penny is the elf and a level 20 archbishop who fights chimeras with a bolo.
Raj is the gremlin and a level 32 sorcerer who fights giants with fireballs.
Sheldon is the ogre and a level 24 paladin who fights harpies with a mace.
Explanation
The first step is to determine the actual skill levels of the five players.
The lowest level is 10, held by the character using a dagger (clue 8)
From (4), the character using a crossbow has an unknown level, let's call it x; the paladin has x+8 and Raj has x+16. Note that x must be greater than 10 and less than or equal to 16, or else x+16 would be greater than 32.
The elf is none of these four players (2, 11, 6), but has a skill level of 10 more than one of them (6). This could be 20, x+10 or x+18.
One skill level has to be 24 (1), two skill levels must have a difference of 4 (7) and one skill level must be twice another (3).
If x=11, then the skill levels are 10, 11, 19, 27 and the elf's is either 20, 21 or 29. None of these is 24; therefore, x isn't 11. Similarly x isn't 12, 13 or 15.
If x=14, then the skill levels are 10, 14, 22, 30 and the elf's is either 20, 24 or 32. This provides a skill level of 24, but for the elf, instead of the ogre; therefore, x isn't 14.
Therefore, x=16 and the skill levels are 10 [dagger], 16 [crossbow], 24 [paladin=ogre], and 32 [Raj]. The elf's skill level must be 20, not 26, in order to satisfy clue (7). This means the elf fights chimeras and Leonard's character uses the crossbow.
Now that we know the skill levels, we can start building a list of the players, starting with the first clue:
Player 1 = Penny
Player 2 = ogre, level 24, paladin
Player 3 = thief
Player 4 = fireballs
Player 5 = orcs
Since the elf has a skill level of 20, the kobold fighter is level 10 (6), and uses the dagger (8). Penny's character doesn't use the dagger (8), and neither does the ogre, who is level 24. Therefore the thief does:
Player 1 = Penny
Player 2 = ogre, level 24, paladin
Player 3 = level 10, thief, kobolds, dagger
Player 4 = fireballs
Player 5 = orcs
Neither Penny's character (12) nor the ogre (4) uses the crossbow [Leonard], so the orc fighter must:
Player 1 = Penny
Player 2 = ogre, level 24, paladin
Player 3 = level 10, thief, kobolds, dagger
Player 4 = fireballs
Player 5 = Leonard, level 16, orcs, crossbow
The ogre can't use the bolo, as no one has level 12 (3), so Penny's character uses the bolo and the thief is the halfling. This leaves the ogre using the mace:
Player 1 = Penny, bolo
Player 2 = ogre, level 24, paladin, mace
Player 3 = halfing, level 10, thief, kobolds, dagger
Player 4 = fireballs
Player 5 = Leonard, level 16, orcs, crossbow
This means that Raj's character, at level 32, must use fireballs, and Penny's character must be the level 20 elf who fights chimeras:
Player 1 = Penny, elf, level 20, chimeras, bolo
Player 2 = ogre, level 24, paladin, mace
Player 3 = halfing, level 10, thief, kobolds, dagger
Player 4 = Raj, level 32, fireballs
Player 5 = Leonard, level 16, orcs, crossbow
The ogre can't use the bolo, as no one has level 12 (3), so Penny's character uses the bolo and the thief is the halfling. This leaves the ogre using the mace:
Player 1 = Penny, bolo
Player 2 = ogre, level 24, paladin, mace
Player 3 = halfing, level 10, thief, kobolds, dagger
Player 4 = fireballs
Player 5 = Leonard, level 16, orcs, crossbow
This means that Raj's character, at level 32, must use fireballs, and Penny's character must be the level 20 elf who fights chimera:
Player 1 = Penny, elf, level 20, chimeras, bolo
Player 2 = ogre, level 24, paladin, mace
Player 3 = halfing, level 10, thief, kobolds, dagger
Player 4 = Raj, level 32, fireballsx+18.
Player 5 = Leonard, level 16, orcs, crossbow
The ogre, who uses a mace, doesn't fight giants (10); therefore the ogre fights harpies, leaving Raj's character as the giant fighter:
Player 1 = Penny, elf, level 20, chimeras, bolo
Player 2 = ogre, level 24, paladin, harpies, mace
Player 3 = halfing, level 10, thief, kobolds, daggerx+18.
Player 4 = Raj, level 32, giants, fireballsx+18.
Player 5 = Leonard, level 16, orcs, crossbowx+18.
The hobbit doesn't fight giants (5), so the hobbit must fight orcs, leaving Raj as the gremlin. The gremlin isn't the archbishop or the merchant, and so the gremlin is the sorcerer:x+18.
Player 1 = Penny, elf, level 20, chimeras, bolo
Player 2 = ogre, level 24, paladin, harpies, mace.
Player 3 = halfing, level 10, thief, kobolds, dagger
Player 4 = Raj, gremlin, level 32, sorcerer, giants, fireballs
Player 5 = Leonard, hobbit, level 16, orcs, crossboe
Leonard's character must be the merchant, rather than the archbishop (7), leaving Penny's character as the archbishop:
Player 1 = Penny, elf, level 20, archbishop, chimeras, bolo
Player 2 = ogre, level 24, paladin, harpies, mace.
Player 3 = halfing, level 10, thief, kobolds, dagger
Player 4 = Raj, gremlin, level 32, sorcerer, giants, fireballs
Player 5 = Leonard, hobbit, level 16, merchant, orcs, crossbow
Howard's character doesn't fight harpies (13), so that falls to Sheldon's ogre, while Howard's halfling fights kobolds. And that completes the list:
Player 1 = Penny, elf, level 20, archbishop, chimeras, bolo
Player 2 = Sheldon, ogre, level 24, paladin, harpies, mace+18.
Player 3 = Howard, halfing, level 10, thief, kobolds, dagger+18.
Player 4 = Raj, gremlin, level 32, sorcerer, giants, fireballs+18.
Player 5 = Leonard, hobbit, level 16, merchant, orcs, crossbow+18.
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