The son of historical painter Johann Jakob Kirchhoff, he was born in Berlin, and educated there. He then taught in various colleges until, in 1865, he was appointed professor of classical philology in the university of Berlin, where he remained for the rest of his life. Kirchhoff's scientific studies covered a wide range in linguistics, antiquities, and Greek epigraphy.
Xenophon, Respublica Atheniensium (On the Athenian Constitution; 3rd ed., 1889)
Über die Entstehungszeit des Herodotischen Geschichtswerkes (2nd ed., 1878)
Thukydides und sein Urkundenmaterial (1895).
The following works are the result of his epigraphical and palaeographical studies:
Die Umbrischen Sprachdenkmäler (1851)
Das Stadtrecht von Bantia (1853), on the tablet discovered in 1790 at Oppido near Banzi, containing a plebiscite relating to the municipal affairs of the ancient Bantia (the Stadtrecht)
Das Gotische Runenalphabet (1852)
Die Fränkischen Runen (1855)
Studien zur Geschichte des Griechischen Alphabets (4th ed., 1887).
The second part of vol. iv. of the Corpus Inscriptionum Graecarum (1859, containing the Christian inscriptions) and vol. i. of the Corpus Inscriptionum Atticarum (1873, containing the inscriptions before 403) with supplements thereto (vol. iv. pts. 13, 1877–1891) are edited by him. From 1860 to 1902, he was in charge of the Inscriptiones Graecae. He edited Hermes (1866–81).