Logic behind the strokes of complex characters
Chinese characters can be broken up into a number of categories, only one of which are pictograms like you described.


象形字, or pictograms, are simple characters like 日, 山, 口 that are visual 
representations of the words that they mean.
指事字, or simple ideograms, are simple characters like 上 and 下 which are visual representations of more abstract concepts, like up or down.
会意字, or compound ideograms, are characters that are combined of multiple parts that represent more abstract meanings, like 休 (person + tree = rest)
假借字, or rebus characters, are characters that had their meanings significantly changed since Classical Chinese and are now used for completely different meanings. Many of these characters were originally pictograms or simple ideograms, but now have different meanings. Examples are 自 which used to mean "nose" but now means "self" and 北 which used to mean "back" but now means "north."
形声字, or phono-semantic compounds, are the most numerous types of characters in modern Chinese. These characters are comprised of two sections - phonetic and semantic. The character you asked about, 吃, is an example. 口 is the semantic section - it tells that the character's meaning has something to do with the mouth. 乞 is the phonetic part - pronounced qi3, it means "to beg" and is used as a rough guide to the pronunciation of the character. These guides aren't 100% accurate, as the characters often predate phonetic changes in the language.


Hopefully this answers your question - eventually, as you delve into further study, you'll learn about radicals and other deconstructions of Chinese characters which will hopefully demystify the system more for you.

You can read more about the classification of Chinese characters http://en.wikipedia.org/wiki/Chinese_characters here.