Swords and crossbow are fine to use for damage, food and drinks (not Witcher potions) are fine to use for healing.
The easiest way of all is to start a new game on Just The Story difficulty and complete the Devil by the Well contract, and the Griffin contract.
I did try doing this on my Death March run, but found that the health leech on the Noonwraith was undoing any damage I could do at that low level. Later on I completed the Shrieker contract outside Crow's Perch using only sword/crossbow/food, but didn't get the achievement to pop. What I did notice was that the sword I was using had 0 uses left of an oil on it, it's possible that that counted as me having used a blade oil, so I would advise using a sword that's never had oil on it - really just completing this on the first two contracts is by far the easiest way.
Just to point out, PurpleFiftyTwo has advised the Noonwraith (Devil by the Well) Contract is possible on Death March:
If you're trying to do the Noonwraith on a death march playthrough, I can confirm it is possible. Wait for her attack, quickly dodge back then get in a couple of quick attacks. Repeat, patiently and carefully. Stay near the boundary. When her 3 clones appear, sprint/roll out of range immediately, and when they disappear (you don't need to kill them if you move away), get straight back at her. This will minimise her health regen, and after several rounds of the above you can eventually prevail.