There are some related tips, but I did not find any one as this.
The sentence:
1) he considers himself a healthy person because he does some sport and neither smokes, drinks nor takes drugs
According the Cambridge Dictionary website (http://dictionary.cambridge.org/grammar/british-grammar/neither-neither-nor-and-not-either), I suppose that it would be something as follows:
2) he considers himself a healthy person because he does some sport and he does not smoke, drink nor does he take drugs.
I was thinking about these other options:
3) he considers himself a healthy person because he does some sport, does not smoke or drink, neither does he take drugs.
4) he considers himself a healthy person because he does some sport, and he does not smoke or drink, neither does he take drugs.
I have no idea which one is correct or sounds better.
EDIT
Thank you everybody! Now I am clear with it.