It depends on if the bearing cups are shot. You just have to pull the axle apart to check. Get on parktool.com for illustrated instructions. You almost certainly need new cones and bearing balls but you may be able to reuse the hub if the cups aren't pitted.
You may be under or over tightening the quick release or the axle may be misadjusted. The way you install the cones and locknuts is absolutely critical. There should be just a touch of wobble in the axle when the quick release is loose but when tightened it should all go away but without excessive pressure or rotating drag. The quick release actually compresses the axle a bit and makes the bearings run tighter.
That all comes later though- for now, you have to take the axle apart and see if the metal is starting to crumble along the little shiny track the bearing balls make where they roll. You have to check the cone on the axle and the cup inside the hub.
A preliminary check would be to remove the wheel and twirl the end of the axle around with your fingers while pressing inward on it, to see if it feels very rough. Most bearing balls, cups, and cones feel a bit irregular even when new, but it shouldn't feel like the metal's started to crumble a bit.
If you find it is starting to pit, there's no saving it because it will only get worse and the crumbly debris in the grease will only accelerate wear.