How to tell which tier of Heir rune to equip

In general, different "types" of enemies, as well as bosses, have weaknesses.

The most common weakness you'll find is for the "Beast" type enemies/bosses. For example, Cleric Beast, Blood-Starved Beast, or the final form of Father Gascoigne are all Beast types bosses.

Beasts receive an extra 20% damage if you use a serrated weapon (Saw Cleaver, Saw Spear) and take extra damage from Fire. In general, this works on Beast-type enemies as well (the werewolves, for example).

A less common enemy-type is Kin. For the most part, it looks like they are weak to Bolt and Blood. Most are weak to Fire and Bleed as well, but not all.

Everything else has weaknesses on a case-by-case basis, though, so just see what works. For a full breakdown of enemies and their weaknesses, see here.

This is a story event. This particular fight is tricky, because you are meant to aid one of them (Eileen) in killing the other (Henryk), but without aggro-ing her either.

This particular event triggers after talking with Eileen The Crow in the Cathedral Ward, and then opening the shortcut to the Grand Cathedral.

In this fight, she will be attempting to kill Henryk (Later found out to be Gascoigne's partner in the Hunt). If you let this play out without intervening, there is a very small chance Eileen will survive (Eileen surviving is the intended outcome). So you are meant to step in and help. The problem is that if you deal too much damage to Eileen, she will aggro on you as well.