The word translated 'fulfill' is the root 'pleroo' in Greek.
In the Septuagint (Greek translation of the OT, from about 250 BC) the word 'pleroo' in Greek translates the Hebrew word 'malei'. It almost always means 'to fill.' But there are other nuances of the word.
Here is the Strongs's definition:
4390. male', maw-lay'; or mala' (Esth. 7 : 5), maw-law'; a prim. root, to fill or (intrans.) be full of, in a wide application (lit. and fig.):--accomplish, confirm, + consecrate, be at an end, be expired, be fenced, fill, fulfil, (be, become, X draw, give in, go) fully (-ly, -ly set, tale), [over-] flow, fulness, furnish, gather (selves, together), presume, replenish, satisfy, set, space, take a [hand-] full, + have wholly.
Here is an example that conforms to the idea that 'malei' (equivalent to 'pleroo' in the Greek) can mean 'to do.' In the following passage the word seems be used in the sense of 'to confirm, establish, authenticate.'
1 Ki 1:13 Go and get thee in unto king David, and say unto him, Didst not thou, my lord, O king, swear unto thine handmaid, saying, Assuredly Solomon thy son shall reign after me, and he shall sit upon my throne? why then doth Adonijah reign? 14 Behold, while thou yet talkest there with the king, I also will come in after thee, and confirm (Heb. malei, Gk. pleroo) thy words.
There is another word in Hebrew often translated as 'fulfil'; it is the word 'kum.' This word is often used by the Sages in the Mishnah and Talmud and means 'to do, to carry out.' From the following passage you can see that 'malei' and 'kum' are nearly synonymous.
Jer 44:25 Thus saith the LORD of hosts, the God of Israel, saying; Ye and your wives have both spoken with your mouths, and fulfilled (malei) with your hand, saying, We will surely perform our vows that we have vowed, to burn incense to the queen of heaven, and to pour out drink offerings unto her: ye will surely accomplish (kum) your vows, and surely perform your vows.
So to translate the Greek 'pleroo' of
Matthew 5:17 as 'end, terminate' is only one possible reading. I believe that the literary/cultural/historical context of this verse lends weight to tranlating 'pleroo' as 'confirm, authenticate' and probably also 'do, keep.'
Finally, this reading is in accord with Jesus' attitude about the Law in his words in Luke 16:17:
But it is easier for heaven and earth to pass away than for one dot of the Law to become void.