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From Proto-Indo-European*h₂r̥-sḱe-ti(“to start, rule”), which appears to be an inchoative formation from *h₂er-(“to fit, put together”), thus “to begin to fit together” > “to start, rule”. Compare Old Armenianարքայ(arkʻay, “king”), which Matzinger considered inherited from Indo-European, but which is generally thought to be ultimately a borrowing from Greek.
Alternatively from a root *h₂ergʰ-(“to begin, rule, command”), perhaps cognate with Lithuanianregė́ti(“to watch, see”). (Can this(+) etymology be sourced?)
The Epic infinitive ἀρχέμεναι(arkhémenai) is attested.
The Doric future middle form ἀρξεῦμαί(arxeûmaí) is attested. Bailly suggests a Doric imperfect middle form ἀρχόμαν(arkhóman), citing the imperfect form ἄρχετο(árkheto) in Pindar. Liddell and Scott suggest a Doric imperfect form ἆρχον(ârkhon), citing the form ἆρχε(ârkhe), also found in Pindar. Bailly further suggests the Doric aorist ἀρξάμην(arxámēn), noting the form ἄρξατο(árxato), which appears in Theocritus.
Bailly suggests an Ionic middle imperfect form ἀρχόμην(arkhómēn), noting the form ἄρχετο(árkheto) in Herodotus. Herodotus also uses the form ἤρχετο(ḗrkheto). Bailly suggests an Ionic perfect participle ἀργμένος(argménos), noting the form ἀργμένης(argménēs), which appears in Herodotus.