Alexa Chung is a proper noun used as a name; in this context treated as a two-word sequence pronounced with attention to the distinct given name and surname. The first name stresses the first syllable, with a/sounds aligning to the standard English /əˈlɛk.sə/ pattern, while the surname is a single-stress morpheme /tʃʌŋ/ or /tʃəŋ/ depending on accent. Overall, it is pronounced as two syllables for “Alexa” followed by one to two for “Chung,” with natural spacing between the names in fluent speech.
Full pronunciation guide