Is it me or someone else paying karma for my sins? According to Buddhism, is it me who is getting punished or rewarded in next life for my deeds in this life?
Neither You nor Anyone else. 

It is not you, because you are dying and reborning with very very short time period all the time we live in Sansara. It is not anyone else because the result taker is a anyway a  part of the series of dying and reborning cycle. 

To understand this you have to understand components of a creature.

maximum number of components that a creature can have are two. They are body and mind. 
minimum number of components that a creature can have is one. That is mind.(Aruupa Brahmas)
(So now at that moment you must realize that there is no soul to a creature.)

In descriptive Abhidhamma, mind is categorized to 4 components as  Vedanā,  Saṃjñā, Saṅkhāra,  Vijñāna.





This image shows the components of the creature without considering time.



Mind is continues flow of born, existence, dying of Naama (Mindlet). mindlet or Naama is the smallest sub component of the mind. 17 mindlets make a mindlet group called Ruupa. 






This image shows how Naamas are born, exist and die with transmitting energy to the next Naama.








This image shows the time intervals of one Naama. Naama's born time=exist time= dying time= t1.
 Time interval between Dead Naama and Born start of Naama = t2. So at that very little time there is no visible mind. At that moment only body is exist. This is the thing called as anathma/anatta.




There can be one mindlet in the mind at once. 
For a time of blink, trillions of mindlets born, exist and die (So one Naama exist less than nanoseconds). 
If creature consists of body and mind combination, it must be considered both to call that as a creature. (At creatures point of view I, my point of view me).
If creature consists of only mind , it must be considered mind to call that as a creature. (At creatures point of view I, my point of view me).
When consider a wheel, engine, chase individually cannot be called as CAR. Likewise, body or mind (When both are there) individually cannot be called as me or I.
This rapid born, existance and deth of Naama can be seen in high meditation levels of vidhardhana meditation.
Now you realize there is no soul or thing that can be called as me because even before a person say me or I there are trillions of born, existence, dying of Naama.
When someone do a Karma, karma is completed in one Ruupa (consecutive 17 Naamas).
After doing the karma that Ruupa is dead. but energy transmitted to next Ruupas.
If creature made karma those Ruupas are dead at 17 Naama's time. So the creature did the karma is dead with Ruupa (But Body didn't dead so actual dead is not visible).
When Karma get punished at proper time the creature's body also can be dead. 
When body dies we call it as real death.
When Karma get punished the Ruupas did the karma is dead and body also can be dead. 
Karma is also get punished to Ruupas. Ruupas feels the senses according to the Karma. 
When Karma is get punished, punished Ruupas are not the Ruupas that done the Karma but Those Ruupas are generated from the same stream of Naama cycle. So Load Buddha have said that the Punished person is Neither You nor Anyone else.