Pursuing one's aspirations can be rendered into English phrases such as 'going after one's dreams' or 'striving towards one's ambition.' Each phrase conveys the idea of actively chasing after one's personal ambitions and goals.

To pursue one's dream

追求自己的夢想：Pursue one's dreams

Chasing after one's dreams can be translated into English as: Pursuing the vision that lies within
It implies striving to achieve personal aspirations and goals with tenacity and diligence, often despite obstacles and setbacks.