I've read the following etymologies
-ī> Directly from the first person perfective ending *-h₂e of PIE. (from Origins of the Greek Verb by Andreas Willi, Pg 8). He states the PIE *-h₂e became -ai in Old Latin, subsequently evolving into -ī.
-isti> Probably from PIE *-s-th₂e . The /s/ aorist marker was attached to the stem, followed by Stative *-th₂e, yielding -isti. Here, the aorist -s-, the origin of the Pluperfect and Future Perfect endings in -r-, did not undergo rhotacization due to the fact that it did not occur intervocalically.
-it> Again, from an old Latin diphthong, in this case -ei. *-e was the PIE perfective third person ending. Old Latin had what Willi calls "primary" -i, from the -i suffixed to the "secondary" endings (-m, -s, -t) to get the "present tense" primary endings of PIE (-mi, -si, -ti). Then, the third person singular marker -t may have been added as -ī > -ei became indistinguishable from the first person.
-imus> again, the use of the -i perfect marker, followed by the regular reflex of the PIE 1st person plural "primary" ending *-mos (ie -mus).
-istis> again, an amalgam of different perfect markers (-i and aorist -s-) followed by the regular present -tis ending. I am not sure of the origin of -tis, since the PIE active indicative 2nd person plural ending is -te. Perhaps the -s arose from analogy with the -s of -mus?
-ere/-erunt> As I was writing this, I found a great answer about -ere/-erunt, specifically, that is better written and clearer than what I had been saying. This answer is only about -ere / -erunt, but contains a clear explanation of the primary -i as well. Here it is:
What is the origin of the 3rd-person plural perfect ending "-ēre"?
As you probably know already, these endings were appended to an already-modified stem. Proto-Italic developed a perfect in -w- (voiced bilabial fricative). This may have from the PIE active past participle marker -wos-. The -w- perfect became -v- in Latin, and became the "regular" perfect, but many other paradigms existed in classical latin, including ablaut (capo/cepi) reduplication (do/dedo) and the long vowel perfect. -v- had an allomorphs of -u- in certain contexts, eg monēo/monuī.
I hope that helps a little bit.