I have understood your problem. The whole string is actually a JSON object, where quote(") is present in the form of " ;.
First replace & quote; it with ". Then evaluate the expression and get the value of any item you want. 
Below working code :)
<script type="text/javaScript">
var str1="{"actor":"100003221104984","target_fbid":"286108458103936","target_profile_id":"100003221104984","type_id":"17","source":"1","assoc_obj_id":"","source_app_id":"2305272732","extra_story_params":[],"content_ti":{"actor":"100003221104984","target_fbid":"286108458103936","target_profile_id":"100003221104984","type_id":"17","source":"1","assoc_obj_id":"","source_app_id":"2305272732","extra_story_params":[],"content_timestamp":"1325711938","check_hash":"892251599922cc58"}}";
var ans=str1.split(""").join("\"");
var obj=eval("(" + ans+ ')');
alert(obj.target_profile_id);
</script>
And see how your actual Data looks like:
{
    "actor": "100003221104984",
    "target_fbid": "286108458103936",
    "target_profile_id": "100003221104984",
    "type_id": "17",
    "source": "1",
    "assoc_obj_id": "",
    "source_app_id": "2305272732",
    "extra_story_params": [],
    "content_ti": {
        "actor": "100003221104984",
        "target_fbid": "286108458103936",
        "target_profile_id": "100003221104984",
        "type_id": "17",
        "source": "1",
        "assoc_obj_id": "",
        "source_app_id": "2305272732",
        "extra_story_params": [],
        "content_timestamp": "1325711938",
        "check_hash": "892251599922cc58"
    }
}