Ideally, you wouldn't need your alarm clock to wake up. You'd get enough sleep and wake up at the same time every day. If that doesn't work for you, however, try creating a mental alarm before you fall asleep at night.
The National Sleep Foundation says:
To stop using an alarm, you need to create a consistent rhythm from day to day. If you go to sleep around the same time every night and, before drifting off, tell yourself when you need to wake up in the morning, you can actually train your body to come to at the right time. But this won't work if you're exhausted. No amount of circadian rhythm training can help you if you are getting less sleep than you need.
Dean Bokhari also shares this same tip on Medium:
Before you go to bed at night, tell your brain, "wake me up at __ a.m." Unless you got drunk the night before, you'll likely start waking up right before the alarm clock startles you awake.
I've done this before and it seems to work. Bokhari says it has something to do with your brain's reticular activating system (RAS), the part of your brain that regulates sleeping and waking as well as attention.
Set up a backup alarm just in case your brain isn't paying attention, though.