1To chase or run after someone/something; also used figuratively for pursuing a goal or dream without giving up.
犬が公園でボールを追いかけている。
The dog is chasing a ball in the park.
警察は逃げた犯人を必死に追いかけた。
The police desperately chased after the fleeing criminal.
彼は俳優になる夢を追いかけて上京した。
He moved to Tokyo, chasing his dream of becoming an actor.
子どもが弟を追いかけて庭を走り回っている。
The child is running around the yard chasing after his little brother.
2Used to mean 'shortly after' or 'immediately following' - only in fixed forms like 追いかけて or 追いかけるように, not as a standalone verb usage.
台風が去った後、追いかけるように大雨が降った。
Right after the typhoon passed, heavy rain followed immediately.
第一便に追いかけて、二便目のメールが届いた。
A second email arrived right on the heels of the first.