Back in 2010, Marvel Studios confirm that they had the right to The Punisher and said : "hope to bring him into the fray shortly." This is how I would introduce him in the Marvel Cinematic Universe.
The tone:
It could be a lower budget movie release from the studio. Now in don't know if Marvel Studios would do a R rated movie. Since it's now own by Disney, I would say no. If it's PG-13, than it need to be push to the max, like Daredevil, True Grit (2010) and live free or die hard. I would not mind no swearing, but a Punisher movie needs blood. Even if it's in the shared universe of Marvel this movie need to have a more dark tone. It could star more light, like Iron Man but after the first act it need to get dark and gritty.
The story:
I think a good way to introduce Frank is to do again is origin story. He was from a Vietnam war veteran, a NYC cop, and in the 2004 movie a FIB agent and former Delta Force. In this new version Frank should start as filed agent for The S.H.I.L.D. More of a Black Ops soldier than a spy. The Movie should be set between Iron Man and The Avengers, so on a 3 year period. The movie could open with Castle in Afghanistan doing mission for the SHIELD, maybe chasing the Ten Rings terrorist group .
He could comment to his partner on how Tony Stark is crazy for coming in the country person. After some trouble with the mission, Frank listen to his wife that is scared for his life, even if she don't know exactly what he does, she know it dangerous and dont like the long absence period especially since he miss ther second child frist bird day. Frank quit SHIELD even if is friend Coulson and Microchip want him to stay. He diced to work for the NYPD.
Just less than 3 year later Frank take is family for a pick-nick. A show down between the two mafia family's at the same place kill all of his family and let Frank very injured. After his recovery he want revenge, but not just on the men who did it, on all the organized crime in New York.
At first he only wear the traditional skull shirt with the black trench coat. He kill some mob's, interrogate others. But after a really big showdown, he need help. Is old friend Mircochip, a weapon developer and technology expert for the SHIELD help him and let him take some things. Frank take a SHIELD uniform and change it in the Punsiher costume.
The bad guys plot need to be elaborate a little more. But SHIELD could disprove of frank method, and it could and whit Furry saying he will watch him and take action if he continue to kill.
The after credit scene could be Frank interrogating some one in an ally, than the shot of the portal opening form The Avengers with the aliens coming out. Frank push the guy his interrogating and pull out a gun.
That how a think The Punisher could be introduce in the MCU. Of course this is not the complete plot of what the movie should be like, it just the introduction of it.