J. Sheidlower, in The F Word, 3rd edition, traces the first sense of it to F. I. Gwaltney's 1954 Heaven & Hell (p.233 where he refers to WWII) "Grimes loves the army and the army's using him for a screw-boy." The first actual use he gives is in J. Blake's 1971 Joint (p. 67, referring to 1954) "They were known as pussyboys, galboys, fuckboys, and all had taken girls' names like Betty, Fifi, Dotty, etc., and were universally referred to as as 'she' and 'her.'" He traces further uses up to 2008, where a work by 50 Cent and D. R. Pledger called Diamond District is cited.
Okay, so I've done some additional research and I've found even earlier usages of the phrase, on the regular.
The rapper "Defari" on his album, The Next Chapter, in 1995, on the song "Big Up" has this verse:
Link to Defari lyrics for the Big Up song
Verse 3: Lyrics are gold plus steel for the raw deal
Feel MC's and DJ's nuff respect y'all real
Low caps I rock on the regular
Brim covers my eyes so ducks can't see their predator
Facefirst when I disperse yet another verse
INTERESTING NOTE:
You can see how he is using the phrase in line 3 to rhyme with line 4, "see their predator".
ADDITIONAL FINDINGS
In 1996 there was a compilation album released by Interscope Records called Insomnia. It included a song called, On The Regular, by Duo. That song has various verses with the phrase: on the regular
HYPOTHESIS
I do remember hearing this term around the early to mid 90's in the US. I was around 20 or so at the time. Of course it's possible that it was used earlier than this but I think it's highly likely that it started around this time, with songs like these, as a result of trying to rhyme words in rap songs.
It's important to note that it became very common, and is even more so to this day, especially in African American and hip hop/rap culture, to create new phraseology and euphemisms. For one it helps create an identity (which is something the early rap culture - especially in the US - really wanted badly) along with the added value of allowing you to rhyme songs that would otherwise be nonsensical.
My position is that this is how we have come about many of the colloquialisms of today. This on the regular phrase is one but there are many, things like: "throw up a deuce" (to show a "peace" symbol), "roll a fatty (or blunt)" (talking about smoking pot), bling (jewelry), rims or 20's (nice wheels on a car), "gimme a 40" (pronounced "foe-tee") to mean 40oz beer (it was common in the "hood" or ghetto for people to buy 40oz alcoholic beverages). There are tons of these types of words and phrases and many variations. "I'm out" (leaving) comes from rap's early "Audi 5000 G", which just meant, "I'm outta hear guys". Audi 5000 was a popular and desired car by that culture at the time. It eventually morphed to simply "Audi" and then finally "out" or "outta" or "outtie".
FINAL THOUGHTS
So, basically, these words are a way to create identity within subcultures and they are also the mechanism to create rhymes for songs.
I would peg the origin of this particular phrase at around 1990-1995, probably occurring on the west coast and/or southern parts of the United States. Defari was a west coast [California] rapper at the time.
Best Answer
Note: It seems pretty unavoidable that any answer is going to have to address both sex and race. Consider yourself warned.
I've found antedatings for both senses. It looks to me like "Becky" started out as a very white name (see also "the Universal name for White Girl"), then became associated with fellatio in 2009. It's hard to say when the "entitled white girl" sense of the word came about since it's hard to say if a fictional white entitled character is called Becky because that's a white name or if it's because it's an entitled white name.
Although I read some articles that claim the origin of the entitled white Becky is Becky Sharp, a character in the 1847 novel Vanity Fair, I don't think I believe this is particularly connected to modern usage other than as evidence that it's a white name. The book doesn't seem to be consciously making a comment on Becky having racial privileges or being racist, which are present in later iterations of the usage.
It is probably significant to mention the white girls from the song Baby Got Back (1992) by Sir Mix-A-Lot. One girl says to the other:
Here there's definitely racism. And possibly entitlement, in the sense that according to Sir Mix-A-Lot "the African-American idea of what was beautiful was shunned".
Urban Dictionary has the following definition for the "white, entitled" sense:
UD also has a number of definitions given starting in 2009 that mention it as a term for fellatio. The relevance of that date is that in 2009 an artist named Plies released a song called Becky which starts off with the following:
One entry in UD speculates on why "Becky" was chosen for the song:
It's significant to note that there's another earlier mention of a white girl named Becky in Position of Power (2005), but this is brief and is clearly a name, not a term for the act:
Another important instance of the epithet Becky is Beyoncé's Sorry (2016):
Diana Gordon, who "teamed with Beyoncé to pen tracks [...] including 'Sorry'" in an interview in Entertainment Weekly dismissed the claims that "Becky with the good hair" refers to any specific person.
See also Merriam Webster, which gives a very similar answer to mine (although I didn't see it until after I basically finished writing this answer).