Instead of sliding the unlock bar to unlock, slide on the text notification, that should open the text.
Also, go into your Notification settings and see what is in the group called In Notification Center. If you see something in there about Weather (besides the one called Weather Widget, which is the weather icon that came on the homescreen of your iphone, installed as stock by Apple and does not send notifications but lets you see your local weather when you pull down the Notification screen). In you find an additional weather app in your notification center, open that and then slide off the button on the Notification Center line. HTH