Note: I will be updating this shortly with more information (battle strategy, weaknesses, etc). If you have tips, please tell me!
There are a grand total of 26 Witcher Contracts to complete. Every single one is listed as a Contract in your quest log, so there should be no confusion.
Progression, Difficulty and Level have no effect on this achievement. So you can wait until after you've finished the game on Death March!, set the difficulty down, and then blow up the low level enemies in seconds. If you're struggling with any of the fights, this is the way to go.
Reports in the comments say that you can choose to not kill enemies when given the chance and still get the achievement. The only thing you must kill is Woodland Beast (which isn't listed as a Contract, though it may be - there's a separate achievement for it).
(3) Contract: Devil by the Well (Notice Board in White Orchard) (Tip: Use Yrden to deal more damage. I didn't and used Quen instead on Death March - it was slower, but safer)
Contract: The Griffin from the Highlands (Optional according to comments below, though I cannot confirm - Part of Secondary Quest: Master Armorers in Velen - Speak with Fergus, the Blacksmith in Crow's Perch, to start)
(10) Contract: Jenny O' the Woods (Notice Board in Midcopse) (Tip: Use Yrden to deal more damage. When she splits into three, use your crossbow to 1-shot the parts)
(10) Contract: The Merry Widow (Notice Board in Lindenvale)
(33) Contract: Missing Brother (Notice Board at the Inn of the Crossroads)
(20) Contract: Mysterious Tracks (Notice Board in Lindenvale)
(22) Contract: The Mystery of the Byways Murders (Notice Board in Oreton)
(7) Contract: Patrol Gone Missing (Notice Board across from quartermaster's tent at the Niflgaardian Central Army Camp)
(23) Contract: Phantom of the Trade Route (Speak with a refugee in Benek in far East Velen)
(8) Contract: Shrieker (Notice Board in Crow's Perch)
(12) Contract: Swamp Thing (Villager to the west of The Orphans of Crookback Bog) (Tip: Use Aard to knock the Foglet out of the mists so you can attack it)
(6) Contract: Woodland Beast (Various Notice Boards in Novigrad and Northern Velen)
(14) Contract: The Apiarian Phantom (Notice Board in Beanston)
(35) Contract: The Creature from the Oxenfurt Forest (Notice Board in Oxenfurt)
(15) Contract: Deadly Delights (Notice Board in Hierarch Square)
(24) Contract: Doors Slamming Shut (Notice Board in Hierarch Square)
(13) Contract: An Elusive Thief (Notice Board in Hierarch Square)
(25) Contract: Lord of the Wood (Notice Board at the Cunny of the Goose Inn)
(26) Contract: The Oxenfurt Drunk (Notice Board near the Oxenfurt Port)
(16) Contract: The White Lady (Notice Board between Portside and Glory Gates, south of Novigrad's walls)
(28) Contract: Dragon (Notice Board in Fyresdal)
(19) Contract: Here Comes the Groom (Notice Board in Svorlag)
(29) Contract: Missing Son (Notice Board in Rannvaig)
(18) Contract: Muire D'yaeblen (Notice Board near wharf of Kaer Trolde Harbor)
(17) Contract: The Phantom of Eldberg (Notice Board by longhouse in northwest Arinbjorn)
(16) Contract: Strange Beast (Notice Board in Larvik)