。
She finally achieved her goal.
Literal
She [topic-は] finally goal [object-を] achieved.
目的を達成する is the standard set collocation for 'achieve a goal.' Both 目的 (mokuteki, 'goal, purpose') and 達成 (tassei, 'attainment, accomplishment') are heavy Sino-Japanese compounds — natural in formal/written register but a touch stiff for casual conversation. Native speakers in casual settings would more likely use 目標を達成する or simply ようやくできた ('finally got it done'). ついに here marks the long arc of effort culminating in success, with the slight emotional weight that distinguishes it from the more neutral ようやく.