About a year after Michael jackson released Billie Jean, a follow-up
'response' song was released by a lady...this song sounded vaguely
like Billie Jean (same rhythm) and was sung as a response to MJ's
song. Vaguely remember the lyrics that included 'I'm billie jean...'.

Do you know the name of the song, or the singer?

The song "I'm Billie Jean," released in 1982, was a real oddity. It
was a near-clone of Michael Jackson's original song, both in music and
in lyrics. The female singer was never identified, and the song was
released only on a 12-inch EP titled "Michael." The EP was not in
general distribution to the public, but it got some airplay by radio
DJs and clubs.

"The appearance of two Michael Jackson -related 12-inch EPs of
mysterious origin - both called Michael - bearing the caveat 'These
are weird renditions of weirder songs originally recorded by the
weirdest artist' - was very intriguing... the Billie Jean answer song
I'm Billie Jean (the kid was his son) is also entertaining."

Now Toronto
http://www.nowtoronto.com/issues/2003-07-03/music_perlichspicks.php

Here you can listen to a clip from "I'm Billie Jean":

Wax Records
http://www.waxrecords.com/store/getTracks.asp?albumID=761&trackID=2203

"I'm Billie Jean" is a rather rare item that was released only as a
12-inch vinyl EP. You can purchase "I'm Billie Jean" here:

Musicstack
http://www.musicstack.com/item.cgi?item=9867474&find=i'm+billie+jean

A different response to "Billie Jean" was called "Superstar (I'm
Billie Jean and I'm Mad as Hell.") It was released in 1983, and was
sung by Lydia Murdock:

"Billie Jean" was made into 'Superstar (I'm Billie Jean and I'm Mad as
hell)' by Lydia Murdock."

i80s Forums
http://i80s.com/forums/archive/index.php/t-1895

Here you'll find the lyrics to Lydia Murdock's "Superstar":

Bide et Musique
http://www.bide-et-musique.com/song/1940.html

You can purchase Lydia Murdock's "Superstar" here:

Musicstack
http://www.musicstack.com/tsearch/lydia_murdock/superstar

(NOTE: The Musicstack site sometimes suffers from server overload; if
this happens, and you can't get on the site, just try again a few
minutes later.)

Yet another song that is a response to "Billie Jean" is "Still Your
Son," sung by Toni Estes. The song is written from the perspective of
the fictitious "Billie Jean," who reminds the father of her child,
"The kid is still your son." Michael Jackson's song, as you'll recall,
contained the refrain "The kid is not my son."

"STILL YOUR SON... seems like a response to Michael Jackson's 'Billie
Jean': "Why you wanna run from me?/Like I'm the police/Ever since I
had your baby/It's like you don't know me/ You said you'd be there for
me/ Now you wanna play dumb/But the kid is still your son.' That last
line is delivered the exact same way MJ delivered it in the
aforementioned classic."

ePinions
http://www.epinions.com/content_116912524932

Here you'll find the lyrics to "Still Your Son":

Lyrics Cafe
http://www.lyricscafe.com/e/estes_toni/005.htm

"Still Your Son" appears on Toni Estes's CD "Two Eleven," which can be
purchased here:

Half/eBay
http://half.ebay.com/cat/buy/prod.cgi?cpid=4942625&domain_id=1876&meta_id=2

Google Web Search: "i'm billie jean"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22i%27m+billie+jean

Google Web Search: "still your son" "toni estes" "billie jean"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22still+your+son%22+%22toni+estes%22+%22billie+jean

I hope this information is useful. If anything is unclear, or if a
link doesn't work for you, please request clarification; I'll gladly
offer further assistance before you rate my answer.

Best regards,
pinkfreud

Lydia Murdoch did the answer track to Billie Jean called SUPERSTAR. 
Released in September 1983 it got to No 9 in the UK chart.