As far as we can tell, all writing systems with independent origins started out as logographs that where stylized representations of the words, and then they started to change by adapting the rebus principle--basically, puns based on homonyms rather than their abstract representation. Egyptian, Sumerian, Mayan, and Japanese all made translations thence into syllabaries, whereas Chinese remained primarily logographic. Egyptian was borrowed into Phoenician as an abjad (like an alphabet, but no vowels), whence the Greeks added vowels as separate letters, and the Indians readded vowels by modifying the base characters (known as an abiguda), and further complicated the system by stacking the consonant clusters on top of each other.
Chinese can represent phonetic systems by a similar rebus principle. One crucial difference is that syllabaries and abigudas tend to work on having one character per consonant-vowel pair, which works poorly on languages like English where consonant clusters can get atrocious (strangsts: 1 vowel, 8 consonants). Many languages tend to limit themselves to CV or CVC (which can be handled by special rules like C₁V₁ + C₂V₁ really means CVC, e.g., Mayan), so it's not so much a problem unless you're trying to transliterate completely different languages. Of course, Latin script has its own problems in transliteration: it's atonal, it relies on voiced/voiceless distinctions (Chinese uses aspirated/unaspirated primarily!), and good luck if you have languages that have four sounds we'd analyze as "h."
I'll point out that Chinese script has been used to write other languages--most notably Japanese, Korean, and Mongolian, none of which are related to the Chinese languages.
Another problem with Chinese is that, because you can write the same sound many different ways, you can make mere phonetic transcription a political deal. A transcription might happen upon the character for "small" or "stupid" or "backwards," which might not go over well if it's someone's personal name.