This word is always spelled this way, with one and only one '-h-': character. It is not uncommon to see it written with an '-h-' after each '-c-'. There should only be an '-h-' after the first. The same is true of the verb to characterize, sometimes spelled characterise - see -ise - -ize) (~ 'to portray as having a certain character'), and the adjective and noun characteristic. (The word caricature has no '-h-' at all.)
Characteristics is one of the 117 mis-spellings listed as 'Common difficulties' in the section on 'Spelling' within 'Writing' in UEfAP.