The Reclassify Tool will convert your raster into a format which allows for Attributes to be added to the Raster itself.
If i have understood your question, you are looking to add the elevation ranges to the attribute table so that you can use this information in perhaps the legend of a description of what the newly reclassified values mean.
Right-click the reclassified raster and you will see the "Open Attribute Table", alternatively you can hold the "CTRL" key and double click the layer in the ToC.
Add a field and call it "Range", make it text.
You can either start editing and manually change each of the ranges t othe description you want...or...
You can create a table containing the reclassified value and the description you want, and then perform a table join between the table and raster - using the reclassified values as the linking field. You can then copy the joined data into the field you just created.