Ms. Tanaka likes Mr. Yamada a lot, so she chases him every day.
OMG! Koichi left for Japan without his passport. Can you run after him and give him this passport?
I saw Fugu following a pod of killer whales.
You know that 追う is "to follow." So, you're following something or someone, or something similar. Just think of the かける as you being all "cocky" about it. You're not just following this person, you're chasing them, with the intent to catch them. So, this word is to chase and to pursue. Scary!
The reading comes from 追う as well as the kanji reading you learned too.