When someone does [罪]{つみ}を[着]{き}せる, if there is no 罪, it's impossible to do 着せる, so 罪 is supposed to exist. However, it's possible that 罪を着せる is used to express actions related with prospective 罪, as mentioned by narutoさん in the answer.
The expression 罪を着せる does not specify what someone actually does. So, the actual action of 罪を着せる could be lying, remaining silent, blaming someone else without proof, destroying real evidences, inventing fake evidences or things like that.
But whatever s/he does, the point of 罪を着せる is that s/he "puts the blame on someone who is innocent" or "sets someone up to take the fall."
It doesn't matter much who is guilty of the crime actually. Almost anyone can be the subject of the idiom 罪を着せる. For example, let's say, there are four people here and a crime occurred. No one, except the culprit and the witness, knows who did that.
A: is guilty of the crime and keeps it secret.
B: is the witness and knows A did that.
C: is innocent and does not know anything about the crime.
D: is also innocent and does not know anything about the crime.
「AはDに罪を着せた。」
「BはDに罪を着せた。」
「CはDに罪を着せた。」
These are all grammatical and valid usage of 罪を着せる.
The synonyms of 罪を着せる are
「[罪]{つみ}を[被]{かぶ}せる」
「[濡]{ぬ}れ[衣]{ぎぬ}を[着]{き}せる」
etc.
If you want to express "deceiving" nuance rather than "blaming" nuance, 「仕立て上げる」 can be more suitable, I think.
「たまたまそこにいる[大学生]{だいがくせい}を、[犯人]{はんにん}に[仕]{し}[立]{た}て[上]{あ}げる。」
「いじめっ[子]{こ}が、いじめられっ子を、[悪者]{わるもの}に仕立てあげた。」
These examples are too sad if really happened. But both sentences are grammatical, and both「犯人に仕立てあげる」 and 「悪者に仕立て上げる」 are common usage of 仕立て上げる.
In some situations, 「はめる」 and 「[罠]{わな}にかける」 can be used to describe a similar nuance, as mentioned by user4092さん in the comment. 「[陥]{おとしい}れる」 can also imply a similar meaning. These expressions mainly imply "deceive and drive someone into an awful situation." So I guess that 罪を着せる can be one way of はめる, 罠にかける, or 陥れる, but it's not the only way to achieve them.