Push them in a Manchester canal when drunk.
I’m not sure why you’d fake a DUI, when you could just claim that you weren’t paying attention and had an “accident.” Especially if the intended victim is on a bicycle - in the US, there are many cases where an inattentive driver hits and kills a cyclist, and doesn’t even get prosecuted for anything. Happened a few years ago a couple of blocks from my office - cyclist (college student) got hit in broad daylight and died, driver admitted she was looking down at her phone, no prosecution.
It will me despair for the English language too! 
Aye, there’s the rub.
Fentanyl. The ME would know the person died of a Fentanyl overdose, but the stuff is killing so many people who aren’t traditional drug abusers that it would be very hard to prove murder.
One of the reasons Fentanyl is killing so many is because it is an opioid that is 40 times stronger than Heroin, yet it can be dried and pressed into pill form. Some cartels are importing the stuff disguised as Oxycontin, so innocent people think they’re buying some oxy for the opiod addiction they got after surgery, and they take Fentanyl by accident and it kills them dead.
Put an unmarked bottle of Fentanyl pressed into Oxycontin pill form on the person’s bed stand, and no one would have trouble believing that someone might have purchased a bottle of what they thought was Oxycontin on the black market and overdosed themselves. This is common enough that at least it might be enough to insert reasonable doubt into the prosecution’s case if they ever caught you and prosecuted.
Well, if you’re familiar with the show, Quincy is a self-righteous crusader for social justice–Wiki:
There would be no moral outrage if it was just an “accident” and Quincy couldn’t get up on his soapbox.
P.S.–I wonder what social irresponsibility punk rock was responsible for?
I vaguely recall reading that the KGB used to stuff snow or ice down someone’s throat to suffocate them. The residual body heat was enough to melt the ice, leaving no evidence (assuming there were no defensive wounds, etc.)
Those darn kids these days! I never! Harrumph! 
Quincy’s sponsors thought it was the root of all evils that have befallen civilized life.
In the US autopsies are rare now, happening in about 5% of deaths. A death by heart attack in anyone over the age of 50 is probably not considered suspicious. If you want to discount the Church Committee feel free.
An aside regarding that link–I couldn’t get past the first few paragraphs of the page without thinking the author fell down the evolutionary intelligence tree backwards, hitting every branch on his way down to the bottom.
That was just beyond moronic.
I had a chem prof in college that maintained that if you fed someone enough heavy water they would die and a lab would not be able to tell you why.
Doesn’t sound practical but hey, if it works.
Wiki sez the stuff is biologically toxic, but it’s real impractical to use as a poison on humans. Heavy water - Wikipedia
The problem is you need a lot of it. Easy enough to get in a nuclear plant, but hard to keep the unwitting victim from taking in regular water too & diluting your poison. OTOH, if you totally control the victim’s environment so they’d be drinking pure heavy water for days and days to build up a lethal dose, why not just whack him over the head with a 2x4 and be done with it?
This sounds like Evil Overlord mistake #13 or so.
dasmoocher hit close to the plan that I was going to post. If your victim’s habits allow for it, stage a hit and run. Without witnesses or other evidence, there’s no way to determine it wasn’t just an accident.
Put them in a baby swing and they will die from appedecidis
Fingerprints on their back, brusing from being pushed, hitting the ground faster than if they just fell
And you buddy Christopher Walken doesn’t rat you out.
Unless, Quincy, M.E. decides to investigate!
Seems you have the death certified over the phone and then have him embalmed shortly there after…
Except hit and run where someone dies is still a serious crime even if it was an accident. If someone is killed in a hit and run the police will make serious effort to find the driver, including things like using any flakes of paint from the car on the victims body to identify the model. And most people don’t regularly walk on the road in places that you can guarantee will have no witnesses, unless they live in the middle of nowhere.
But the OP is:
Is there actually a way to kill someone and leave no sign of it being a murder?
Serious accident: maybe–yes?
Murder? Uncertain.