I can't recall ever hearing a person described as "feigned" or "false." Please don't use phrases like "feigned person" or "false person" -- it sounds weird.
"False" can describe actions or feelings. It is very commonly used in the phrase "false modesty." (He showed false modesty.) People also talk about false praise, false encouragement, false sympathy, and false piety.
"Feigned" is almost always used as a past tense verb, NOT an adjective. The most common use in this context is "feigned interest." (He feigned interest in my story, but I know he wasn't really listening.)
I see that "contrived" is suggested above but, again, this word almost always describes actions, not people. Some examples: contrived outrage, a contrived show of pity.
Some good adjectives you can use to describe this kind of person are: insincere, untrustworthy, unreliable, hypocritical, dishonest, disingenuous. Each one has a slightly different meaning, but they all go fine in a sentence like: He is a ___ person. or He is ____.
You can also say, "He is a phony." (NOT "He is a phony person.") This is very colloquial and informal.