Murder level requirements:
(These are cumulative, so for example to meet the requirements for mdr3 you must meet all requirements for mdr1 and mdr2)
1 - 20 Ruins kills (not counting First Froggit)
2 - Kill Toriel
3 - Kill Doggo
4 - Kill Dogamy and/or Dogaressa
5 - Kill Greater Dog
6 - Kill or have already killed Snowdrake
7 - 16 Snowdin Kills (not counting Doggo, Dogi, or Greater Dog)
8 - Kill Papyrus
9 - Kill Shyren
10 - Kill Glad Dummy
11 - 18 Waterfall Kills
12 - Kill Undyne the Undying
13 - Kill Royal Guards, or spare them after killing any other monster
14 - Kill Muffet, or spare her after killing any other monster
15 - 40 Hotland Kills
16 - Kill Mettaton, alongside not having spared/fled the following: RG01, RG02, Glad Dummy, Greater Dog, Lesser Dog, Dogamy, Monster Kid, Doggo, Papyrus (also cannot kill Papyrus while murder level is 6 or lower in 1.001+), Shyren, Muffet, Toriel