Accidentally killed eileen the crow. Can I still get the hunter rune

Enemies can now contain your blood echoes, so to get them back you need to kill that enemy.

I have just started the game myself and haven't noticed this yet, but after researching it appears this is actually a new mechanic in the game. It looks like the enemy has to be near where you died, and can be seen to have glowing eyes (see Reddit).

One poster says:

Yeah it actually says this in the manual. Something like 'your blood echoes will be on the ground unless an enemy gets to them first... which will make them have glowing blue eyes... kill them to regain your lost blood echoes'.

I believe you are referring to the "Approval" gesture. You can get this gesture by following Eileen The Crow's questline.

• To start the questline you must meet Eileen in Central Yharnam, she can be accessed through the sewers and the room she is hidden in is hidden behind breakable objects. By speaking to her you will have also started the questline.
• Her next location is in the Cathedral Ward, she is waiting just to the left of the front exit from the lamp. When speaking to her she will alert the player that she is hunting a hunter at the Tomb of Odeon.
• At this point the lamp at the Tomb of Odeon would be disabled, and the player must backtrack from the Cathedral Ward to the tomb. Eileen's quarry will be waiting, and he will attack you. At this point Eileen will aid you to kill the hunter. As a reward for helping her kill the hunter you will get the Approval gesture. Note that she can be killed in the encounter, and hitting her may trigger her to be hostile towards you.

More information regarding the questline: http://bloodborne.wiki.fextralife.com/Eileen+the+Crow