Sir Isaac Newton led the foundation of his famous laws of motion during the 17th Century but at that time SI system hadn't existed. So in which units did he define force? Did he define it in some other units or was later redefined by scientists who made the SI system?
If you read the article On the Concept of Force: How Understanding its History can Improve Physics Teaching you will realise that Newton's ideas about force are not the same as those of today. Indeed Newton thought that there were different types of force$^1$.
No unit of force was proposed until 1873, when the dyne was introduced. Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces cancelled each other out. The strength of any force was expressed in terms of an equivalent gravitational attraction, with weights or masses measured in grains.
$^1$ I have added an amplification as a result of a comment by @terry-s.
What @terry-s has written is certainly true as explained in The - Stanford Encyclopedia of Philosophy, Newton’s Philosophiae Naturalis Principia Mathematica, 4. “Definitions” and absolute space, time, and motion
Newton distinguishes among three ways of quantifying centripetal forces: the absolute quantity, which corresponds to what we would call the field strength of a central force field; the accelerative quantity, which “is the measure of this force that is proportional to the acceleration generated in a given time;” and the motive quantity, which is the measure of the force proportional to what we would call the change in linear momentum in a given time.
In The Principia. Preceded By A Guide To Newton's Principia By I. Bernard Cohen: Mathematical Principles Of Natural Philosophy there is the following paragraph:
Another significant novelty of the second edition was the introduction of a conclusion to the great work, the celebrated General Scholium that appears at the conclusion of book 3. The original edition ended rather abruptly with a discussion of the orbits of comets, a topic making up about a third of book 3. Newton had at first essayed a conclusion, but later changed his mind. His intentions were revealed in 1962 by A. Rupert Hall and Marie Boas Hall, who published the original drafts. In these texts, Newton shows that he intended to conclude the Principia with a discussion of the forces between the particles of matter, but then thought better of introducing so controversial a topic. While preparing the second edition, Newton thought once again of an essay on "the attraction of the small particles of bodies," but on "second thought" he chose "rather to add but one short Paragraph about that part of Philosophy." The conclusion he finally produced is the celebrated General Scholium, with its oft-quoted slogan "Hypotheses non fingo." This General Scholium ends with a paragraph about a "spirit" which has certain physical properties, but whose laws have not as yet been determined by experiment. Again thanks to the researches of A. Rupert Hall and Marie Boas Hall, we now know that while composing this paragraph, Newton was thinking about the new phenomena of electricity.
The forces on particles of matter is elaborated on:
In a draft of the General Scholium (ULC MS Add. 3995, fols. 351-352, published by Rupert and Marie Hall), Newton outlined, but did not give details concerning, the attraction among “very small particles,” explaining that it is “of the electrical kind.” He then lists briefly some of the properties of what he calls “the electric spirit.” Since that publication in 1962, there has been no doubt that the “spirit” mentioned in the final paragraph of the General Scholium is electrical.
In Newton's time, it was not general practice to define physical units. There were in general no equations, no constants of proportionality.
This seems to be explained in terms of the practice of expressing physical relations by proportion, so that -- if one thinks in terms of modern analogies -- any constants of proportionality just cancel out. A previous topic of discussion illustrates this, I think. So I would suggest looking at Who first derived $a =v^2/r$ , especially at part (b)(3) of the answer. This shows an English translation of Huygens' description of various physical relationships, all having to do with centrifugal force (the 5th paragraph of (b)(3) is the start of the translation). The relationships are expressed in terms of proportions, not equations, and the extract shows how the style used then did not require units nor constants of proportionality. I hope that helps.
Physics is not mathematics and so units are always required whether explicitly as today or implicitly as in Newtoms day.
It appears in Newtons time force was generally expressed in terms of an equivalent weight. This is natural quality to use even if the unit of such is left undetermined.