Formal, often used in business or official contexts. Literally 'as quickly as possible'.
Please reply as soon as possible.
If a problem occurs, we will deal with it as soon as possible.
More casual and conversational than 可及的速やかに, which is formal and often written.
Also formal, but 至急 is a single word meaning 'urgently', while 可及的速やかに is a longer, more explicit phrase.
A set phrase composed of 可及的 (kakyūteki, 'as much as possible') + 速やかに (sumiyaka ni, 'quickly'). The word 可及的 is a Sino-Japanese compound from 可 (ka, 'possible') + 及 (kyū, 'reach') + 的 (teki, adjectival suffix).