This is quite the tricky question. I am going to assume that the character you describe is your protagonist.
#How To
How To
The character you describe is best classified as a "sociopath":
a person with a personality disorder manifesting itself in extreme antisocial attitudes and behavior and a lack of conscience
Rather than the more easily concealed "psychopath":
a person suffering from chronic mental disorder with abnormal or violent social behavior; an unstable and aggressive person.
As such, the only way to hide this information from the reader is to hide her inner monologue (If you had a psychopath, or were writing a story about passing on, you could have an utter disconnect between her perception and the real world). Doing so with her as the narrator, or having a narrator privy to her thoughts, would require her inner monologue to be outright missing in some sections where she is deceiving someone.
If you were to change her from sociopath to psychopath, then you could hide her lying under her own delusions of "doing the right thing," however that is neither as interesting nor what you asked for.
Thus, the best option to hide her insanity is to have another character on the ground serving as the narrator. This character would have to be unfamiliar with your protagonist, so they can't see through her act. If you are really up for the challenge, you could have your narrator for each of your sections, then they can be disposed of in an indirect way by the end of the section (ex: your narrator for the "alien invasion" section could be tripped when fighting in the last battle, or something). Generally, they should be disposed of (either the story moves past them or they die) after, but not immediately after, they narrate your protagonist acting odd (not that they necessarily have to even note that something is odd).
#Should You?
Should You?
If you integrate small hints correctly, this can be a very fun character to experience. This gives your story re-readability, and subtle enough hints will lead to needing multiple re-reads to get everything, just as cool background details and small lines in movies and TV give them re-watchability.
If, however, you have her acting perfectly normal (not not always having that perfect veneer, but either always having it on or maybe saying some really out-of-place things), then it will feel like an incredibly cheap plot twist.
As a result, whether or not you should is entirely a matter of your appraisal of your writing abilities and your personal vision.