Your raster is not only lacking proper CRS information, but also the right extent in map units (degrees). Usually it can be extracted from the file's metadata (but sometimes QGIS fails on that), or the information is only given on the download site.
So you have to tell QGIS that the extent is between +/-180°/90° with gdalwarp (see How do I make gdalwarp set target extents to -180 -90 180 90?How do I make gdalwarp set target extents to -180 -90 180 90?) or gdal_translate -a_ullr -180 90 180 -90 -a_srs EPSG:4326
.
Both can be run on the command line, or from the Raster
menu.