In the context you gave, about would be better:
Money is the least thing I should care about.
I agree with cornbreadninja about the word least, though. Last sounds more natural:
Money is the last thing I should care about.
Care of is not used as a verb phrase. Likewise with care. You cannot care of something, and you also cannot care something. A legitimate verb phrase using care of is take care of, which means protect/meet the needs of. Also, as FumbleFingers says below, care of is possible when care is a noun.
As a side note, if you ever see the common phrase care for, bear in mind that its meaning is different from care about, especially when used with a negative.
If you don't care for something, it means you don't like/prefer it.
If you do care for something, it means you take care of it.