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From what I understand Khosrow I, being the leader of the Persian Empire of 532 CE, had the title/name. What is the origin of the name Khosrow?

According to Google Translate, "khosrow" in Persian means King, but I was unable to determine if that meaning came first or Khosrow I did or Julius Caesar did.

No. The Persian name is a derivation or descendant of the legendary Kay Khosrow. Looking at the list of name bearers Khosrow reveals that the name is in much longer use than 532 CE.

Variants of the name کیخسرو‎

Husrav, Xusro, Khusro, Khosrau, Khusrau, Chusrau, Khosro, Khosru, Khosrow or Khusraw. In Greek it is sometimes rendered C(h)osroes or Osroes. Arabic it's Kisra (Kisrā) and Turkish Hüsrev. Note that in Kaisar is the Turkish word for Caesar.

The meaning of Khosrow would be "of good standing" or as Wikipedia puts it:

The name Kay Khosrow derives from Avestan Kauui Haosrauuah,2 meaning "he who has good fame".

And on Wiktionary:

Etymology

From Middle Persian 𐭧𐭥𐭮𐭫𐭥𐭣𐭩‎ (hwslwdy /Husrōy/), [Book Pahlavi needed] (hwslwb' /Husrōy, Husrav/), from Old Median *Husrava, from Proto-Iranian *Hhuĉráwah (“reknown, famous”).

Compare the Encyclopaedia Iranica entry:

KAYĀNIĀN vii. Kauui Haosrauuah, Kay Husrōy, Kay Ḵosrow

The name Haosrauuah is a vriddi formation of *husrauuah “he who has good fame” and ought to mean “good fame” by itself. The later forms, in fact, seem to be descended from *hu-srauuah, although shortening of the initial syllable is possible. The Pahlavi form is usually spelled hwslwy, hwslwd, or hwslwb, all of which should probably be read as Husrōy. The Persian form may derive from this, or it may have been remade in analogy with compounds with -sraw. (On the morphological irregularities of the Avestan name, see Humbach and Ichaporia, 1998, p. 137.)

In the Avesta. The Avesta contains more details about Kauui Haosrauuah (Figure 3) than any of the other kauuis, except Vištāspa. His standing epithets are “stallion of the Aryan lands” (arša airiianąm dax́ iiunąm) and xšaθrāi haṇkərəmō “.?. for command,” where haṇkərəmō has not yet been conclusively interpreted (Yašt 5.49, 15.32). It is also transmitted as haṇkərətō “put together” (Yašt 9.21)

Whereas Caesar is very unlikely to be related.

As others have said, the names and titles of Caesar/Kaiser and of Khusrau/Khosrow/Chosroes/Kisra are unrelated. But there is a famous historical example of someone claiming to be related to both Caesar and Khosrow.

The Caliph Yazid III (701-744) recited a poem about his exalted ancestry:

I am the son of Chosroes, my ancestor was Marwan,
Caesar was my grandsire and my grandsire was Khagan.[3]

Which I discuss in post number 5 in this thread: https://historum.com/threads/emperor-maurice-and-khosrow-ii-of-persia.127393/1