Saul came to a city of Amalek - I believe the original should be translated, and Saul came to the city Amalek; their capital being called by the name of their tribe.

And Saul came to a city of Amalek,.... With his army, perhaps the nearest city of it to the land of Israel; though some think that Amalek was the name of the city, and was the metropolis of the nation, and had its name from thence: and laid wait in the valley; which was near the city, to intercept the inhabitants when they should come out against him: or "he contended" (w) as some render it, he fought with them there; the Targum,"he ordered his army,''set them in battle array, or pitched his camp there.
(w) "et certavit Pagninus"; "ut contenderet cum eo", Junius & Tremellius.

Saul came to a city of Amalek--probably their capital.
laid wait in the valley--following the strategic policy of Joshua at Ai (Joshua 8:4).

He then advanced as far as the city of the Amalekites, the situation of which is altogether unknown, and placed an ambush in the valley. ויּרב does not come from ריב, to fight, i.e., to quarrel, not to give battle, but was understood even by the early translators as a contracted form of ויּארב, the Hiphil of ארב. And modern commentators have generally understood it in the same way; but Olshausen (Hebr. Gramm. p. 572) questions the correctness of the reading, and Thenius proposes to alter בּנּחל ויּרב into מלחמה ויּערך. נחל refers to a valley in the neighbourhood of the city of the Amalekites.