PGF1:
On the other end of the scale though, the aftermath of his life is a prime example of being in the shadow of an even greater historical figure, Sir Isaac Newton.
THOUGH is redundant with "On the other end of the scale"
Fame and glory aside though, the results of his work still had a great impact on astronomy as they laid the stepping stones for further developments in the field.
THOUGH is again probably not necessary.
Little is known about his other two siblings unfortunately, other than their respective birth and death years, both having died at a very young age.
I would probably move UNFORTUNATELY to the front.
PGF2:
The lack of financial concerns within his family, and throughout the beginning part of Halley’s career had the biggest influence on his uprising as, “one of the greatest men of his lifetime” (Heckart. 1984, pg 10).
FAMILY, AND doesn't seem right, the comma shouldn't be there, nor should the comma before the "one of the greatest men...."
From an early age his father encouraged Halley to pursue his interest in astronomy and purchased a lot of instruments for him.
HIS FATER ENCOURAGED HALLEY makes it sound like you're speaking of someone other than Halley's father. Perhaps reword it to "From an early age Halley's father encouraged him...."
He was already noted as a very knowledgeable astronomer by that time, with a very fine collection of instruments, which his father kept on providing throughout the years.
Seems odd. This, with, which. Perhaps break into 2 sentences or reword a little so it flows better.
PGF3:
After being in College for a very short time, Halley left for reasons that are not fully known, and departed on a journey to map the southern hemisphere.
No comma after the KNOWN, it's really not necessary. Although it also sounds as if his journey was the reason for leaving, even though you say he left for reasons unknown. I don't know the whole story here, but from reading the sentence it seems contradictory to itself while trying to be explanatory.
The whole trip was funded partly by his father and King Charles II.
I'd probably just say "The trip...." rather than "The whole trip...." WHOLE followed closely by PARTLY always looks funny. 
Or maybe say "The whole trip as funded jointly by his father and King Charles II."
In 1678 Halley had become a graduate of Queen’s College with the help of the King, at the same time becoming a member of the Royal Society, making him one of the youngest people there.
I'd probably restructure this a little. Instead of "King, at the same time" I'd say "King while at the same time...."
PGF4:
The later parts of Halley life became quiet turbulent.
QUITE not QUIET....just like red and read.
In 1694, Halley’s father had died, found ditched on the side of the road.
That just sounds strange. I would definitely rephrase it. Perhaps "In 1694 Halley's father was found dead, ditched on the side of the road." Or "In 1694 Halley's father died and was found ditched on the side of the road."
The reasons for his death were unknown, and the only clue was that his second marriage was a disaster and a financial strain.
WERE unknown or ARE unknown? Watch your tenses.
Flamsteed later on retrieved all of the books published and in, “1715 he…publicly burned them ‘that none might remained to show the ingratitude’ of two of his countrymen” (Moore & Mason, 1984, pg 38).
The comma after AND IN probably shouldn't be there. Perhaps do AND IN 1715, "he publicly...."
Halley now had an Astronomer Royal as his enemy.
Astronomer Royal....haha, sounds like Battle Royal. Just thought it was amusing.
Both of these disasters interrupted Halley’s research until Flamsteed had died and Halley was named the new Astronomer Royal.
HAD DIED? I'd probably ditch the HAD. More tense switching.
The friction that existed between him and her husband prompted Mrs Flamsteed to take all the instruments which where bought by Flamsteed himself.
Ambiguous. "him and her husband" isn't clear enough as to who the HIM is. I'd restructure it a little to make it more clear.
This was a very devastating result and the instruments where never found to date.
WERE not WHERE. However I'd ditch it anyways. "...the instruments have not been found to date." Or "...the instruments were never found." Or something of the sort.
PGF5:
When Halley had first become friends with Flamsteed, he heard of his plans to compile a catalogue of the Northern Hemisphere.
HE...HIS. Ambiguous. "...he heard of Flamsteed's plans...." perhaps?
In 1676 Halley had left Queen’s College and had sailed to St. Helena where he planned to do most of his research. Upon his arrival home in 1678, he had managed to record the positions of 341 stars and observe a transit of Mercury across the Sun's disk. Halley showed both Flamsteed and Robert Hooke, the secretary of the Royal Society, his work, leaving both of them stunned that such a young man could come up with such results. Halley later on published his findings in a book called The Catalog of the Southern Stars.
You keep switching here with "had left" and "had sailed" and "had managed" but then you switch to "showed" and "published". To stay consistent I'd probably ditch the HADs.
PGF6:
When Halley was touring France and Italy, visiting different observatories and astronomers, he saw something that had him fascinated.
Sounds a little odd, I'd maybe say "...that fascinated him." rather than "...had him fascinated." It's just a peculiar way of putting it.
This got Halley interested in them, so he began researching these celestial objects from an observatory he built at home.
Is the comet the focus, or Halley? If Halley is the focus, I'd say "This interested Hally, so he...."
This was a great triumph for the long gone Halley and Newton, since after all, the principles used in the calculations where his.
I'd probably ditch the comma after the AFTER ALL.
PGF7:
It was in fact the persuasion he gave Newton to publish Principia, a book that advanced Newton’s idea of force, and planetary motion.
No comma after FORCE. Only use the comma when there's more than 2 items in a list, otherwise AND is just fine on its own.
Newton being a shy and quiet person never publicly told anyone about any of it though, and was not planning to publish his results in the future either.
A little odd. Probably say "Newton, being a shy and quiet person, had not yet told anyone about any of it and was not planning...."
Later one when the book became a big hit, he made most of the money back from sales.
Later ON not ONE
In the end, what it all comes down to though is Halley’s persuasive techniques and financial capabilities being the main reasons why Newton had managed to publish the book.
I'd probably ditch the THOUGH here again
PGF8:
Halley had other contributions to the scientific world and even though they were not as important as the three mentioned above, they still had their impact on society.
Ya know those commas I've been complaining about, add one. "...scientific world, and even though...."
Today Halley is thought of as a first person to studied scientific geophysics.
"a first person" ?? The, perhaps? Or "one of the first people...."?
Halley also worked with many other scientists such Robert Hooke, and contributed his ideas where he could.
such AS Robert Hooke
PGF9:
Halley’s three major contributions to the scientific world are important to many people, and still affect us today in some way.
I'd ditch the comma.
His chart of the Southern Hemisphere played a big role back in the day, since travel below the equator was now made possible since, “it would then be safe for navigation, commerce and exploration” (Heckart, pg 10).
"back in the day" sounds very informal. I'd also ditch the comma after "made possible since ...."
Halley’s discovery of comets, specifically the work he had done predicting the return of Halley’s Comet was partially an important discovery for Halley himself, since it is arguably his only token to fame.
"...comets, specifically the work he had done predicting the return of Halley's Comet, was particularly an important discovery for himself since it is arguably his only token to fame." Note PARTIALLY changed to PARTICULARLY and the extra comma after Comet. However calling it Halley's Comet in the same sentence describing his prediction is a little odd, not sure how to get around it though.
Many people have even said that, “it is the greatest work on natural science ever done by a man” (Heckart, pg 55).
Many people have even said "it is the...."
It was also not until the 20th century, during Albert Einstein’s time that Newton’s principles that been slightly adjusted.
"It was also not until the 20th century, during Albert Einstein's time, that...."
Sir William C.D. Dampier, a historian of the sciences, had said that Newton showed that, “heavenly bodies worked in one gigantic mathematical harmony.”
Ditch the comma.
Other than aforementioned comments in italics, it's pretty good.
Hope this helps.
b