How do I replace my bulging battery please?

Fast action is needed. A swelling battery will continue to "grow and at a rather alarming rate. The more it bulges, the more non-recoverable damage is done to the internal components like the trackpad, keyboard, and logic board.

Could I replace it myself?

Realistically, only for MacBook Pro sub-models made before 2013 NOT having Retina displays. I cannot in good conscience recommend that any end user attempt a battery replacement on the Retina models. It is highly invasive, requires the use of some really gnarly chemicals to unglue it, and is best left to the professional. I am comfortable getting inside almost any computer, but I will not attempt to replace the battery in a Retina MacBook Pro.

"About this Mac..."from your Apple menu will show your sub-model:

Contacting AppleCare as QuickPost has suggested is the only way to find out if Apple will still service the battery in that model.

If it is supported, this article contains the US price list (parts and labor) for battery service.

Mac Repair - Official Apple Support

With stores closed, you will have to arrange for a ship-in service.

Hello,

Turn off your MacBook, unplug it from power, and stop using it. Swollen batteries can be very dangerous. You will need to call Apple for assistance.

Contact AppleCare

If the laptop is a non-Retina model, then it is possible for a user to replace the battery fairly easily. Check OWC for a battery and instructions. Keep in mind that the quality of third party batteries is poor even from a reputable vendor such as OWC.

https://www.macsales.com

If you have a Retina model, then you should definitely get an official Apple repair.

I highly recommend an official Apple repair since an original Apple battery tends to be much more reliable, but I understand the virus crisis makes this difficult since Apple Stores outside of China are closed this week. An independent AASP may be open though.