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To convert a vector layer or an Openlayers.Geometry object to a GeoJSON string, you should use the OpenLayers.Format.GeoJSON.write function:

var geoJSON = new OpenLayers.Format.GeoJSON(),    geoJSONText = geoJSON.write(geometryObject);

Note that you should be able to pass your object to this function, since (according to documentation) it accepts an OpenLayers.Feature.Vector as well as a OpenLayers.Geometry or an array of features.

Conversely, when you’ve got a GeoJSON string, you can convert that back to an object using the OpenLayers.Format.GeoJSON.read function:

var geometry = geoJSON.read(geoJSONText, 'Geometry');

The second parameter lets you indicate which type of object you’d like returned. Read the docs linked to for more information.

Also, take a look at this demo for a more extensive example. (View the source of the page to see how they’re doing it).