Identify the symptoms. Closely examine when, where and how they occur. Attempts to repair the display could be a waste of time and money if the problem hasn't been properly identified. If the problem is caused by software, a simple system restart could make it go away. Make sure the problem is recurring before assuming that it needs fixing.
Test the symptoms under various conditions to see if they are consistent. For example, a laptop display that turns black could simply be turning off due to the energy saver settings. Visual artifacts, incorrect sizing and flickering could be attributed to a bad display driver. Some things to try are booting into safe mode (press F8 at system start), updating the video drivers and physically adjusting the screen.
Check for physical damage. If the screen is cracked or shows any signs of damage, it's pretty safe to say that is the cause. The next step is to figure out if replacing the screen will fix it, or is the problem deeper than that. A bad inverter can cause symptoms similar to that of a broken screen. The only way to tell is to replace either one and see if the problem is resolved.
Hook the laptop up to an external monitor. This is a way to check if the problem is hardware- or software-related. The display will usually show up on the second monitor without any additional configuration. If it doesn't, you will have to find out if the video drivers have that capability. The feature to look for is "cloning" or "dual display" properties.
Replace the graphics processing unit (GPU). Laptops manufactured in the last year or two are being made with desktop-size GPUs that can be replaced or upgraded. Unfortunately, in older laptops this is impossible because the graphics card is fused to the motherboard and cannot be replaced. In this instance, you may be able to find a used replacement from an online merchant or auction.