Why Lash has been killing Inhumans?

Initially, it was just something innate...when he killed he felt better.

Wikia

Garner's body went through Terrigenesis, where his body became cocooned and every cell in his body was changed. As his body became freed, Garner felt as though he had completely changed and had an incredible desire to track down and be near any other Inhumans, although he did not understand why.

"I didn't know why, it was intense. I had this new desire, I had a hunger on my skin. It wasn't until I finally met one that my body took over and I, I lashed out and I felt relief."

^{Andrew Garner - AOS, Chaos Theory}

Using Jiaying's book, Garner was able to track one of the Inhumans. Feeling an incredible desire to be near him, Garner approached and suddenly transformed into a monstrous grey-skinned beast. Garner's instincts took over and he killed the Inhuman by ripping a hole in his chest.

The Inhumans believe that every Inhuman was born for a reason, it seems likely (and I believe stated) that Lash was the counter-weight to Hive's influence...hence his ability to "cure" Daisy.

Part of this was as a result of Andrew Garner's ability to control himself when transformed into Lash which enabled him to use his Inhuman abilities without killing.

Note: The Marvel TV version of Lash is significantly different to the comic version where Lash embarked on a mission to find all the individuals affected and judge for himself whether they were worthy to live with their new abilities.

Wikipedia reference to Entertainment Weekly article in which...

Executive producer Jed Whedon explains that the show's adaptation of the character is different from the comics, but some elements are inspired from the source material.

When the Kree realized that Hive had the ability to assume control of all Inhumans, they made the decision to create Lash, an Inhuman who would hunt down and kill others like him. This would prevent Hive from creating an army from other Inhumans, which nearly happened in late Season Three. The Kree also insured that Lash would be the one Inhuman incapable of falling under Hive's control.

Lash was never created to save Agent Daisy Johnson's life. That was Andrew Garner allowing his feelings and connections to Daisy take over, while he was Lash.