。
She ran as fast as she could to catch up with the others.
Literal
She [topic-は] other [genitive-の] people [to-に] catch-up [in-order-to-ために] possible-as-far-as fast ran.
Three patterns. 追いつく ('catch up to') takes its target with に; the verb is built from 追う ('chase') + つく ('attach, reach'). ~ために is the purposive 'in order to.' できるかぎり ('as much as possible / to the extent one can') is a fixed adverbial — できる (potential of する) + かぎり ('limit, extent'). 速く is the adverbial form of 速い ('fast'); not to be confused with the homophonous 早く ('quickly, early'), which is more about timeliness than speed.