No I do not think he's dead. We didn't actually see him die, which is strange since the important character deaths are shown (see Billy and Bob). Also during the shot of the machine when it's about to explode Hopper isn't standing next to it, which begs the question: where did he go?

Many do believe Hopper is the American prisoner - or they did at first. But there is another theory that's doing the rounds that's more likely: he's in the Upside Down and the prisoner is Dr. Brenner. There a couple of shots that focus on the portal of the Upside Down and Hopper also looks at it. And then there's the line at the end of the letter: "keep the door open three inches". That might have a deeper meaning. He probably went there thinking that he might have a chance of surviving (and El could get him out with her powers) whereas if he just stayed where he was there was absolutely no chance of him making it out alive as he'd be vaporized.