Google added the lock screen widgets to Android from Jelly Bean 4.2.2 and if your device is running 4.2.2 or up, then you can view weather on lock screen of you Android device, by doing as follows:
1. The first thing you need to do, is installing an Android Weather App which supports lock screen widgets. What I personally use is Transparent clock & weather, which adds a transparent weather forecast on lock screen.
2. To add lock screen widget, make sure your device is locked and then press power button to turn the screen on
3. Now Swipe right as much as you see the Plus icon (In other words, you need to pass the current lock screen widgets to be able to add another on an empty spot)
4. On the list of available lock screen widgets, Tap Transparent Clock to add it on the lock screen
5. Now, you should see the time as well as the current temperature and weather condition
** Note that, if you are using Android 4.4 (Android KitKat), you need to enable Lock Screen Widgets first. Learn How to Enable Lock Screen Widgets on KitKat.