。
She tried to run as fast as she could.
Literal
She [topic-は] [as much as possible-できるかぎり] fast [tried to-run-走ろうとした].
できるかぎり is just the kana spelling of できる限り (the kanji 限 reads かぎり) — same word, different orthography. Kana spelling is more common in casual or quick prose; the kanji form looks slightly more formal. The volitional 走ろう plus とした gives the past-tense 'tried to run' — the difference between this and a plain 走った is that 走ろうとした includes the attempt aspect, leaving open whether the run actually happened or how far it got.