To take advantage of another's crisis
To take advantage of another's crisis or weakness by using blackmail or some other form of coercion
Negative
他突然对记者大发雷霆，称他们是一帮乘人之危的家伙 | Tā túrán duì jìzhě dàfāléitíng, chēng tāmen shì yī bāng chéngrénzhīwēi de jiāhuo
He suddenly raged at the journalists, calling them a pack of vultures [who took advantage of others' misfortune]
[Source]
19世纪末，西方列强乘人之危，胁迫清王朝签订了不少不平等条约 | 19 shìjìmò, xīfāng lièqiáng chéngrénzhīwēi, xiépò qīng wángcháo qiāndìng le bù shào bù píngděng tiáoyuē
At the end of the 19th century, western powers took advantage of the Qing government's weakness to force it to sign many unfair treaties
[Source]
To hit a man while he's down
None
This is a new HSK chengyu
This chengyu story requires a basic understanding of Han dynasty government administrative divisions. I've included a link to an English-language Wikipedia page that gives a more extensive overview, but effectively, this is the breakdown: the biggest region is a province (in this case, it's translated as 州 (zhōu)), and within the province there are commanderies (郡, jùn), and within the commanderies there are counties (县, xiàn). The administration of a province's commanderies is overseen by an "inspector" (刺史, cìshǐ), and a commandery was under a "grand administrator" (太守, tàishǒu)
《后汉书》(Hòu Hàn Shū, "History of the Eastern Han") by 范晔 (Fàn Yè)
During the Eastern Han Dynasty (东汉 | dōng hàn | 25 - 220) there was a brilliant and upright commandery official named 盖勋 (Gài Xūn). The commandery in which he worked, 汉阳 (Hàn Yáng), was a part of the province of 凉州 (Liáng Zhōu), and the inspector of the province, 梁鹄 (Liáng Hú), was one of his good friends.
Now, the administrators of 凉州 were very corrupt, and the corruption ran from them all the way up to the imperial court. This became a problem when one of 粱鹄's non-corrupt subordinates, 苏正和 (Sū Zhèng Hé), began to investigate corruption of the 武威 (Wǔ Wēi) commandery grand administrator, who was also under 粱鹄's jurisdiction. Worried that he and the court would be implicated by 苏正和's findings, 粱鹄 began to think of killing him to prevent the truth from getting out. He soon turned to 盖勋 to discuss his plan.
As it turns out, 盖勋 and 苏正和 were arch rivals, but when he heard his friend's plan, he was staunchly against it. It was dishonest and not benevolent, he explained, to take advantage of the situation and kill of 苏正和.
In the end, 粱鹄 did not have 苏正和 killed.