Dictionaries call this word a verb, but it doesn't seem to behave like any other verb in the English language. Another question on this site calls it a “conjoined pronoun-verb combination”, which seems nearer the mark to me. What is the reasoning behind the dictionaries’ decision to label methinks as a verb?
Probably the easiest way to understand the syntax of methinks (in the present tense, that is; methought is its past tense) is by viewing it as saying “it thinks to me that”, where thinks here actually means seems, and that leading me is a “dative of interest”. Curiously, this is not the only verb to do this. Its synonym meseems (along with meseemeth also in the present and meseemed in the past tense) works exactly the same way, and for just the same reasons, too.
Like its synonym meseems, methinks is an archaic impersonal verb with two odd characteristics: despite its impersonality, it stands without a dummy it, and it includes a prefix me serving as an old “dative of interest”. That is the sort of dative that occurs when someone says “What happened to your old beater car?” and you say “What, that old thing? It broke down on me one time too many, so I junked it.”
Both meseems and methinks were originally written as two separate words: me seems and me thinks in the modern spelling, and in Old English variously me þinceþ, me þinceð, me ðincþ, me ðincð, and many others besides. Over the course of time, the original construction became obscure, many variant forms were seen, and they became fused into single words each.
The original ameaning of the Old English verb þyncan, ðyncan is not the one we use for think today, and indeed, this sense of think is preserved in modern use only in methinks alone. That original and now obsolete meaning of think was to seem or appear, which is exactly what it still means here. The seem meaning of think grew rarer after 1250, and disappeared some time after 1400 except in this frozen impersonal expression, methinks. For several centuries its written form varied wildly, because this sort of dative fell out of productive use and people became confused about what was going on and how it all worked.
So methinks is a real verb, albeit a curious one. It is in the present tense; its past tense form is methought. Its OED entry ponders the question of its curious syntax, which it calls “obscure”.
The set of forms provided is incredibly long, and contains several entries it thinks may have been transcription errors.
The OED2 has this extended note on its etymology:
It then goes on separately document in the present tense all of me thinketh, methinks, methink, and my thinks(s, and in the past tense all of methought, methoughts, my thought.
The OED3 has an even longer etymological note, of which this is just the beginning:
The reason it doesn't behave like any other verb is that there is no other verb like it — or at least, no Modern English verb.
It’s an Old English verb which means “it seems to me,” and although it survives to the present (OED has a 1991 citation), it’s listed as “archaic, poetic and regional”.
It’s also listed as an impersonal verb (a verb with no subject), used with a subordinate clause or parenthetically:
Dictionaries list it as a verb because that’s what it is.
M-W and YD both label it as an impersonal verb. Apparently, when listing parts of speech, some dictionaries don't make the distinction between a verb and an impersonal verb. Methinks that’s because so few verbs are inherently and exclusively impersonal, but it’s hard to say for sure.