Is it difficult to get accepted to a university as a international student and native English speaker?

It’s not difficult to get into a Korean university as an international student, but you do need some level of Korean.
Most Korean Universities will require a level 3 or higher in TOPIK (Test of Proficiency in Korean) for undergrads and TOPIK level 4 or higher for graduate programs.
 
Some programs taught exclusively in English do exist and these only require you to take the TOEFL test (no Korean), but even then you will definitely have some lectures in Korean. Thus it’s suggested that you do know some level of Korean before you come here.
 
If you’re starting from scratch it’ll probably take 9~12 months of learning intensively to reach level 3 on TOPIK. Please refer to this resource to learn more about studying in Korea.