That doesn't look abnormally bulged to me. Those lines you see on the top are pressed into the capacitor casing during manufacturing. This creates a weak point so that if the pressure in the capacitor exceeds safe amounts, the top of the casing will split open and vent out instead of the capacitor turning into a miniature rocket with a loud bang.
You should measure the capacitor (capacitance, and also ESR if you have an ESR meter) to determine its health, as there don't seem to be any signs of physical distress.
Don't forget to check voltages! That means all power supply outputs for proper voltage and low ripple. Also the LED backlight driver output voltage.
All that being said, a non-working backlight is more often than not caused by bad LEDs in the backlight. However, it's not always easy to determine if the LED array is faulty or the driver circuit, because many drivers will stop outputting voltage if there is an issue with the LEDs causing the current and/or voltage to be outside of the programmed tolerances, so the lack of output voltage from the driver does not necessarily indicate a problem with the driver.
You can test to see if you can measure a brief voltage spike on the LED driver output when turning on the TV. If there is, then most likely the LEDs are at fault. Note that if your multi-meter is too slow, it might not be able to "catch" the driver briefly turning on.
Note also that the LEDs are normally arranged in series-connected strings, meaning the voltage required to light them up can be pretty high (up to a few hundred volts on some TVs). This means testing the backlight by powering it with a common bench power supply isn't likely to be feasible.