Faking your death should have a really dire reason for it. Just disappearing is much easier and safer. But I suppose the steps aren't that different.

Find a country with relaxed visa requirements. Anything where you can get married easily and obtain permanent residency will work. Solo travel to a nearby country. Find some kind of remote tourist attraction where getting lost seems plausible. Tell your friends or family you will go there. But instead of actually going there you take a land route to your final destination. Once you entered the country find someone to get married to to get permanent residency and working rights. Take their name. This step sounds harder than it is, plenty of people are willing to marry someone they just met. Don't forget you are only doing this for the visa not to actually have a fulfilled family life. So you don't have to love or even like them,

You are now in a country no one even thinks about looking for you with a new name. It wouldn't work if you are running from certain government agencies that are acively searching for you but for most other cases this should be easy enough.

You can take this step even further. For example Argentina offers you a second nationality/passport as soon as you have a Argentinian spouse. So you could repeat the steps with your new Argentinian name/passport. I am sure you will find some Argentinian willing to marry you and then let you disappear for a couple of thousands USD.