You can't add a layer package to a web map. This will not change. What you can do is add a shapefile or CSV directly to a web map. From the arcgis.com map viewer, click the Add menu button and add a layer from a file. You can upload a shapefile with roughly 1000 features in it (it depends on the geometry of the features). You'll be able to symbolize the shapefile from the web map.
If you want to upload larger datasets, then your option is to purchase a subscription to ArcGIS Online. With a subscription, you can have ArcGIS Online host your data, creating web services out of them.
http://www.esri.com/software/arcgis/arcgisonline/features/organizations.html
hope this helps,
Mike