What is the difference between a god of murder and a god of war?
Is it scale, or just semantics*? You could say that one is the god of murder and the other is the god of mass murder. Murder has very negative connotations, but war, war is glorious. Your god of murder needs an image change, something to make the concept more acceptable.
Perhaps he's the god of assassins, though this is probably a side line of the god of death. The sailors get a lot of gods of the sea, the sea is a very dangerous place to be. There are also gods of fertility and farming for the more landbound peasantry. There are gods of just about every trade and assassination is as dangerous a trade as any.
There are endless gods of death, everybody has one (or more), after all, everybody dies. It's just a matter of encouraging said god to take the person sooner rather than later.
Maybe a god of vengeance, Adrestia (also known as Nemesis) was known to accompany her father Ares to war. Remember that Ares was himself the god of bloody violence in war:
"overwhelming, insatiable in battle, destructive, and man-slaughtering." His sons Phobos (Fear) and Deimos (Terror) and his lover, or sister, Enyo (Discord) accompanied him on his war chariot.
Athena was the goddess of the just, controlled, tactical war. Politicians might call on Athena when sending the army to war, but the soldiers know that on the field they'll find only Ares. Place your offerings as appropriate and pray that he's on your side.
Perhaps this is a side role of the god of justice, execution is after all a standard acceptable method of punishment in most religions. Though perhaps you want Adikia the goddess of injustice to aid you, because you know you're in the wrong but you still want to be rid of your rival.
*It's semantics, a legal technicality. The difference between judicial execution and a revenge killing is the judicial part, the bit where society approves the action. War could be said to be government sanctioned mass murder. It's just point of view.
This could also go the other way, the negative connotations of murder used against a god.
He wasn't always a god of murder
The problem is that society has turned against this old god and the old traditions. Some new god has come along and now society disapproves of the occasional human sacrifice the old god demanded. Some stand by the old ways and make the sacrifices, but they're considered to worship the god of murder, not by themselves, unless they've taken it as a badge of pride, but by others. Though like the Thuggee they're likely to be cracked down on by the new order if they sacrifice too many people.
Estimates of the total number of victims vary widely, since no reliable source confirms the length of the Thugs' existence. According to the Guinness Book of Records, the Thuggee cult was responsible for approximately two million deaths; British historian Mike Dash said that they killed a total of 50,000 people over an estimated 150 years. Political scientist David C. Rapoport estimated that 500,000 people were killed by the Thugs, making them the most destructive terrorist group in history. According to other estimates, they murdered one million people.
Whether it's human sacrifice, gladiators (to the death), duelling (to the death of course) or trial by combat (need I say it?), a god who encouraged/demanded these things could become known by outsiders as a god of murder.
This is classic us and them propaganda.
- Our great god who loves and protects us, your god of murder who loves human sacrifice.
- Our great god of war who aids us in our glorious battles, your savage god of bloody murder who loves nothing more than a field of corpses and keeps you in a constant state of war. Which way the god falls in this depends on where you stand.
Though perhaps like Ares and others, the god is both aid and terror even to their own side.
On the other hand
Some men just want to watch the world burn