Or HaChaim on Numbers 24:20/אור החיים על במדבר כ״ד:כ׳

1 ראשית גוים עמלק ואחריתו עדי אבד, "Amalek was the first of the nations, but its end will be utter destruction." The word "the first" refers to the first nation Israel has been commanded to destroy (compare Exodus 17,14) where G'd orders the Israelites to record this commandment in a book (before the Torah had been revealed at Mount Sinai) and where G'd promises to wipe out the memory of Amalek. This occurred shortly after the Exodus when Amalek attacked the Israelites in no-man's land without provocation. At that time, the Amalekites left the battlefield and went home. Later on, in the days of Saul, the Israelites almost wiped out Amalek completely except for their king. In the days of Mordechai the Jews killed over 75.000 of their enemies, presumably mostly people descended from Amalek. When Bileam refers to אחריתו, "the end of Amalek," he refers to the messianic era when the Messiah will wipe them out completely.

2Another way of explaining this verse may be as follows. The word ראשית does not describe something chronological, but is derived from ראש, head, i.e. "first in importance." There was a time when the people of Amalek ranked very high amongst the nations. Nonetheless, the time would come when it would be utterly destroyed. It is part of the everlasting glory of Israel that it will be the nation which destroys Amalek. Bileam continues in verse 24 by describing other nations which will experience destruction at the hands of third nations, such as Tzim at the hands of the Kittim, etc. It will be to Israel's advantage that these various nations will be destroyed.