Certainly possible.
You need to go deeper into a specification though. Does your doorbell already have different stored tones and all you have to do is decide which one is selected or do you have to generate the tones yourself.
There are already many 'jingle generator' tone devices on the market that have inputs you connect to ground/supply to select the tone you want. If you use one of those all you have to do is make your remote control produce the appropriate high/low signals to drive the generator. You can use devices like the Holtek HT12 to do that.
If you want to be able to produce your own tones, sending the sequence itself to the doorbell it gets a bit more complicated because you have to send and store the sequence in the doorbell circuit. You will need an MCU to do that. If you have WiFi, something like an ESP8266 can do it and be programmed using an network connected device. If you want to do it without a network connection, a small PIC would be an obvious solution. You would need to think about how the tones were entered and how the remote link works as well though.
Brian.