Fertilizer first, weed killer after, as the weed killer has to remain on the weeds to kill them. Also, make sure it is weed killer, like Weed-B-Gon, not an overall vegitation killer, like Roundup. Also, do not apply the weed killer till the weeds are actually growing and have leafed out, as it is a killer, not a preventer, and must land on weeds to kill them. A preventer would also inhibit grass growth. Spray on a non-windy day, when rain is not in the forecast. Wear long pants, gloves, regular shoes, not sandals.
ETA, if you had granular, growing weeds must be damp for the grandules to adhere to them in order to kill them. Again, no rain after.