Which one is right "Think bad of someone" or "Think badly of someone"?

Certainly,

think badly of

is grammatical and idiomatic here. Macmillan has:

think badly of someone PHRASE

to have a bad opinion of someone or something

Nobody will think badly of you if you fail.

It considers this idiomatic because the adverb badly might usually be interpreted in the 'needs improvement' sense: He spoke badly.

think bad of

is also used {Google Ngrams}, though much less often; it is possibly modeled on the more clearly acceptable think ill of, which ODO wisely lists under 'phrases' rather than a particular POS section for 'ill':

Phrases

2 speak (or think) ill of

Say (or think) something critical about (someone).

Some would doubtless label 'think bad of' incorrect, judging it to be an unjustified flattening of 'badly' to 'bad'. But the usage note given at AHDEL needs careful consideration:

Usage Note: Bad is often used as an adverb in sentences such as His tooth ached so bad he could not sleep. This usage is common in informal speech but is widely regarded as unacceptable in formal writing. In our 2009 survey, 72 percent of the Usage Panel rejected the sentence just quoted.

In any case, 'bad' might be considered a noun in 'think bad of', as in 'desire good for'.

"To think bad" and "to think badly" are both correct depending on the intended meaning. "To think badly" means that one's thinking processes are faulty or defective, as in the statement, "You're thinking badly or not very clearly about the consequences of your actions." The statement, "I feel very bad for the family" means that you feel sad or empathic for the family.

Note the following examples:

The dog smells bad/The dog smells badly.

In the first example, "The dog smells bad," the meaning of the sentence is "The dog stinks." In the second example, "The dog smells badly," the meaning of the sentence is that the dog has a poor sense of smell, that the dog is no bloodhound!

In the following examples, "I feel bad"/"I feel badly," the first sentence, "I feel bad" means that you are sick or that you do not feel very well or that you feel sorry or sad for something. The second sentence, "I feel badly" means that you have poor feeling or a numbness in your fingers.

Thus, in the sentence, "I thought badly of him," means that you were wrong in thinking of "him" as you did. The sentence, "I thought bad of him" means that you thought bad things about him.

In most cases, when the senses are involved, the adjective, rather than the adverb, will be required.

Sorry Michael I do not think you are correct.

"To think bad of..." and "To think badly of..." are two different things:

"Bad": Adjective - of poor quality or a low standard.

"Badly": Adverb - in an unsatisfactory, inadequate, or unsuccessful way, or used to emphasise the seriousness of an unpleasant event or action.

To think badly or "poorly" of someone or something means to hold the thing in low regard or low esteem. It is not normally used literally to indicate defective mental processes.

The dog smells bad/The dog smells badly.

The dog smells bad is an informal usage of the word bad, to suggest the dog has an unpleasant odour. "The dog smells badly" is an informal way of saying that the dog's nose does not work very well.

"I feel bad"/"I feel badly"

Does not mean the things described.

"I feel bad!" is an informal way of saying "I feel ill." or unwell.

"I feel badly [about what has just happened]." Indicates a feeling of regret about an unpleasant event or action.