Reddit mentions of Belkin MiXiT Home and Travel Wall Charger with USB Port - 1 AMP / 5 Watt (Black) - F8J017ttBLK

Honestly, if you have to ask this (and certainly this generally), you should probably think twice before attempting it. Keep in mind that faulty wiring and poor circuit design can easily lead to fires, and these happen all the time, even with "professional" Chinese USB wall-warts.

I highly, highly, highly recommend that you don't attempt to build this from discrete parts (ie transformers, rectifiers, voltage regulators, etc) - I'm an EE, and I wouldn't even consider doing this myself if I wasn't planning on putting 15+ hours of design in just to ensure reliability and safety, as well as another couple hours of testing and verification, ESPECIALLY if I was giving it to someone else. Improper design can, at worst, cause a serious fire (due to thermal failure or internal short circuiting) or electrocute the user with 110v mains due to improper isolation, and, at best, simply not work properly or destroy the device due to poor power regulation.

All that said, Apple has a unique charge scheme for their devices that require placing voltages on the USB data lines to signal the maximum charge current that the device can take. Most Android (and other) devices are signaled that a dedicated charger is present by shorting the data lines together. Between more nuanced things like this and the safety issues I mentioned earlier, your safest bet is to buy and utilize an off-the-shelf USB adapter.

I suggest you buy an inexpensive charger like this and find a way to work it into the clock. Although the form factor might not be completely ideal for you, your chance of making anything smaller yourself are slim to none. Try to keep the case on it if you can, but if it needs to come off in order to fit, make sure you take it apart carefully (a little hot air and prying always works), and then insulate the entire device from both itself and other things using electrical tape.

Then, plug it in to one of these. You shouldn't be trying to remove the existing metal tabs from the charger (or soldering / attaching bare wires to them, either), so you need a plug. The output of that will then have to go into some quick-connect tabs that will let you crimp in this wire. The other end of that wire will then get cleanly soldered to the safest access point from the 120v going into the clock electronics - after which point the wire needs to be strapped down and strain relieved, then both contact points cleaned and coated in hot glue.

Yeah, it's an ugly hack, but it's the safest thing you're gonna be able to get done - anything else would require a lot of reading and learning to ensure that you aren't going to shock someone or start a fire.

Can you compare something like a Belkin or like a Samsung one?
As long as its 1A at 5V, it shouldnt matter right?