How people hear it
Usually if you say it without a pause before the and (or a comma in writing), like this:
It is best to not play video games and read a book before you sleep.
then it would mean "don't do both of these activities together (or don't play video games and then immediately read a book) before you go to sleep."
Usually if you say it with a pause before the and (or with a comma in writing):
It is best to not play video games, and read a book before you sleep.
then it means two separate propositions: (1) It is best not to play video games, ever. (2) Read a book before you fall asleep.
There is no official rule, of course. And in this case, people will probably hear the second version even if you omit the pause (or comma). The reason is, playing video games and reading a book don't combine into an activity you could do before going to sleep.
As it stands, the sentence seems strange because combining those two activities without a pause doesn't really make sense, and recommending against playing video games ever seems strange, too. Why would someone recommend against playing video games ever, and tell me to read a book before going to sleep, all in the same breath? So, a fluent speaker will probably think that you misspoke, and that you meant to say:
It is best to read a book before you go to sleep, not [to] play video games.
Ways to say it clearly
You can use gerunds to make parallel constructions to help the listener connect the phrases correctly:
Reading a book is best before you go to sleep, not playing video games.
Still more ordinary is:
You should read a book before going to sleep, not play video games.
Notice the use of both gerund and infinitive to help the listener keep track of what is supposed to modify what. If you said "You should read a book before going to sleep, not playing video games", that would be heard as "You should read a book before you go to sleep, not before you play video games."
This next sentence is even more lucid:
You should read a book before you go to sleep, not play video games.
The word you helps the listener follows this without confusion, because it puts go into the second person. It's not an infinitive, so the listener doesn't group it with read and play.
If you really want to recommend against doing either activity before going to sleep, then you would say:
It is best not to play video games or read a book before you go to sleep.
People who know set theory or use databases like to say that English uses or to mean and here, but they're mistaken. The usual meaning of or in English is to indicate choices. In this context, it means "either choice you make, the sentence will be true."