It is unlikely that there is a generic legal standard for being improper, or wrongful, instead, impropriety would be a function of the particular context. For an investigation by the Dept. of Justice, it would mean exceeding legal authority or being contrary to law. 28 USC Part II gives great detail on what that means. The Attorney General is at the top of the hierarchy within DoJ, but the Director of the FBI also has statutory authority to investigate certain (allegations of) violation of federal law. You can limit the search somewhat by asking what the AG is allowed to do, which is spelled out in 28 USC Ch. 31. 28 USC 509 is a statutory version of "the buck stops here" (with a few exceptions, all functions of DoJ are vested in the AG). The Justice Manual, which is based on 28 CFR Ch. I gives all of the details. An "improper investigation" would be one lacking legal authority, where authority has to go beyond saying "the boss told me to". An investigation can't just be a matter of satisfying curiosity, there has to be a connection to a "legally wrongful act", and specifically federal law, so either a criminal act or a civil wrong. This section of the manual covers attorney authority regarding criminal matters, and JM 9-2.010 describes investigative authority:
authorized to request the appropriate federal investigative agency to
investigate alleged or suspected violations of federal law.
For instance, it would be acting without authority for the DoJ to investigate unpaid California parking tickets by Nancy Pelosi, because there is no violation of federal law, but it might be with authority (thus, proper) to investigate whether Nancy Pelosi threatened a congressional investigation of a San Francisco police officer, if he didn't tear up her parking ticket (I assume there is a federal law against such behavior).
In the context of the tweet quoted in the OP, it is even less clear what it means to be "improper", since there seems to be no allegation that DoJ carried out an improper investigation. 28 USC 511 authorizes and requires that
The Attorney General shall give his advice and opinion on questions of
law when required by the President
There is no "presidential manual" saying what the function of POTUS is: you have to get that from the Constitution (Article II). This includes (§3) that
he shall take care that the laws be faithfully executed
POTUS can consult with the AG to determine if an investigation is a faithful execution of the laws. In the case of a parking ticket investigation, POTUS ordering the AG to conduct an investigate, "the law be damned", would be an improper order because there is no legal authority for the investigation. POTUS inquiring or requesting is proper (the point of having an attorney general is to advise POTUS on the law: it is not "improper" if POTUS doesn't immediately know the law).