Almost. It should be: "I've read the book three times."
"For" is only used when talking about a duration of time. For example:
"I've been reading the book for 3 weeks."
This is talking about a continued action that lasts over a period of 3 weeks. This type of sentence also implies the action is not yet finished.
When talking about the number of occurrences of an action, "for" is never used. Some additional example are:
"I have only been to america once."
"She has seen the movie 4 times already."
"They have played the game together 5 times this week."
I hope this helps! Please let me know if anything is unclear.