Do this in three steps: break the polygons into their component parts, count the overlaps, and convert to raster. This avoids the potentially huge computational cost of separately converting every polygon to a raster and combining those rasters.
Union (in the
Geoprocessing menu) breaks the polygons into their parts.
Unfortunately, each overlap is duplicated in the output: it has one identical copy for each original polygon covering it. Therefore
Dissolve (again in the
Geoprocessing menu) will merge overlapping parts, provided you can find a way to uniquely identify them. Read through the dialog: towards the end, you will have an option to compute "statistics." Choose any field that may have identified the original polygons and ask for a count.
In many cases the combination of polygon area and perimeter will uniquely identify the parts. If not, you can add more geometric properties in additional fields, such as coordinates of the centroid, until you have accumulated enough information to distinguish every feature.
The resulting layer has one feature for each polygon overlap and some kind of "count" field counting the number of overlaps.
Convert that to a raster, using the "count" field for the attributes.
For example, here are some overlapping polygons and their identifiers with the attribute table shown:
After the second step we have one record for each overlapping region along with a count which can already be used to symbolize the amount of overlap:
The rest is easy--and it's just a single rasterization operation.