If the book you read is in the public domain* you should be fine. Otherwise what you are doing is copyright infringement and probably not protected by fair use**.
One of the rights granted to copyright holders is to control derivative works, and transference to different mediums, which is what your recordings would be.
Under US law, whether an instance of copyright infringement is fair use is evaluated on a case-by-case basis, weighing four points:
the purpose and character of one's use
the nature of the copyrighted work
what amount and proportion of the whole work was taken
the effect of the use upon the potential market for, or value of the copyrighted work
In my non-lawyer evaluation, point 1 depends on what you do in your video (unless you monetize your Youtube video, in which case it is likely to be decided against you), but if you are merely reading the book out aloud, it is unlikely to be in your favor (although it may not be against you as an "educational tool").
Point 2 depends on what is being read, with a informative work (e.g. a textbook) being more likely to be fair use than a creative work (e.g. a novel).
Point 3 depends on how much and what proportion of a work you use; since you are presumably reading a whole book, this would most likely be ruled against you.
Point 4 would almost certainly be decided against you, as you are essentially creating an unauthorized audiobook.
In summary, you can read a book aloud. You can record your reading of it for your personal use. You should NOT upload it to Youtube, or other sharing sites.
*Note that different countries have differing rules on when a book enters the public domain, and since the internet crosses borders, multiple rule sets may apply.
**Technically, even if you had fair use protection, it would still be infringing on the copyright, it would however be legal, as fair use is an affirmative defense.