Chasing tThe mMurderer

This is just capitalizing the title. It doesn't really matter, but titles should follow this formatting.

The police was *looking intoere *investigating* the murder for months.

"investigating" is a stronger verb for "looking into." It can be used when police are looking into a murder. I thought I'd share this with you :)

The *gistkey evidence* was the blood over the *handrail*.

Based on what you said in the comment about "gist," then this should be something along the lines of "key evidence," evidence being the clues used in the investigation.

It wasn't from the victim, so who put it there?

and whyWhy was it put there?

Several months *deemed* the tiniestpassed analyzing all the options.

I think "analyzing" is the verb you are looking for instead of "deem."

Maybe, he didn't *take heed* of some important detail.

I would write this as: "He might have overlooked an imporant detail."

Chasing the murderer

The police wasere *looking into* the murder for months.

The *gistclue* was the blood over the *handrail*.

It wasn't from the victim, so who put it there, and why?

The chief policeman was on the *brink* of madnesspolice chief / The chief detective was on the *brink* of madness after several months chasing even the smallest lead.

Maybe, he didn't *take heed* of some important detail.

more natural: Maybe he overlooked some important detail.

The police wasere *looking into* the murder for months.

The police are plural, so were should be used.

The *gist* was the blood over the *handrail*.

This part doesn't exactly make sense. I don't think that gist fits into this sentence.
The gist of the report was that there was blood over the handrail. -> this sentence could work, but not sure if that is what you meant.

It wasn't from the victim, w. Who put it there and why?

and why?

Several months *deemed* the tiniest option.

This doesn't make sense. Deemed does not fit into this sentence.