What is the meaning of "Ya'an Asos Pelishtim bi'Nkamah"?
Rashi: They did with hatred and oppression, like one who takes vengeance on his enemy. Radak - they did evil of vengeance against Yisrael, which is greater than other evil, as if Yisrael did evil to them initially.
Radak citing Targum Yonasan: They did with hatred.
Malbim: Also they did with hatred. They first did vengeance in the days of Yehoyakim - Hashem sent troops of Pelishtim against him (Melachim II, Perek 24). Afterwards, at the time of the Churban, they took vengeance with hatred - "Al Sheloshah Pish'ei Azah v'Al Arba'ah... Al Haglosam Galus Shelemah Lehasgir l'Edom" (Amos 1:6), "u'Vnei Yehudah... Mechartem li'Vnei ha'Yevanim" (Yo'el 4:6). They took vengeance to destroy via eternal hatred.
What is "bi'Sh'at b'Nefesh"?
Radak: It is disgrace. They disgraced them in their Nefeshes, to do evil to them disgracefully.
What is "l'Mashchis Eivas Olam]"?
Radak: This is a noun, like "v'Hodi Nehpach Alai l'Mashchis" (Daniel 10:8). I.e. they did evil to them to destroy them, so there will be eternal hatred between them.