When should I use ''most of the time'' and ''most of the times'' ?
most of the _______ I wake up at 6am, but today I woke up later.
most of the _______ I would choose apple as my favorite fruit, but the peach was so delicious, so I'd pick the latter.
I like to spend most of the _______ reading books than watching TV.
My brother is a lazy person, most of the _______ he is just playing video games at home instead of studying.

Question

Answer

Use 'most of the time' when the meaning is 'usually'. In all your examples, that is the correct option.
You use 'most of the times' when you use the words literally to mean 'most of the occasions'. e.g. Most of the times when I went to that ski field, the carpark was full. 'Most of the times he visited, I wasn't even here'. 
So - times is for much less frequent occurrences - when you could easily count or remember each one.

Answer

"most of the time" in all cases.  "most of the times" would rarely, if ever,  be used

Answer

Most of the time means usually or very often; most of the times is a bit different and more specific, meaning almost every time (when); almost at every single attempt:

Most of the time I go to bed just before midnight.
but
He used to be a very good soccer player. He almost never missed, he scored a goal most of the times when he was in front of the goal. (almost at every attempt of his shooting towards the goal)