。
She tried hard to express herself well.
Literal
She [topic-は] self [object-を] well express [volitional-よう] [quotative-と] endeavored.
自分をうまく表現しよう (trying to express herself well) uses the volitional form of 表現する (to express). と努めた uses 努める (つとめる, to endeavor/strive) — a formal, dignified verb for making effort. うまく (skillfully/well) targets the quality of expression. The sentence suggests deliberate, purposeful effort at self-expression — perhaps in a speech or interview.