With être + adjective (or a modified adjective), you would use c'est if it is impersonal (i.e., it) and Il/Elle if it is personal (he/she). http://french.about.com/library/weekly/aa032500.htm
'parfait' maps directly into english as 'perfect'. The '-ment' suffix in French corresponds with the '-ly' suffix in English, so the most literal English translation of 'parfaitement' is 'perfectly'. Substituting a different English word for parfaitement is a matter of colloquial preference depending on the context.
For "C'est parfaitement faux" you could say "It's perfectly false," but the implied meaning there's no way for the matter in question to be true, so saying "It's absolutely false" better conveys the intended meaning.
Another example would be "Ils sont parfaits inconnu" which would translate best into English as "They are complete strangers."