An amoeba is an enormously complicated organism. So the answer of how a single-celled amoeba arose is that it developed from a long line of other, simpler organism. But I suspect that that’s not what you’re asking.
I suspect that what you’re really asking is how the first archa appeared. That’s a more difficult question, and we don’t know most of the details. But in broad outline, it seems to be something like this:
1: The pre-biotic Earth had a variety of simple chemicals on it, such as water, methane, carbon dioxide, and ammonia. These chemicals are the most common form for elements such as carbon, hydrogen, oxygen, and nitrogen (which happen to be the primary elements which make up living things).
2: The action of lightning, ultraviolet light, heat, and other energy sources on these simple chemicals caused them to form into more complicated (though still fairly simple, in the grand scheme of things) chemicals such as amino acids and ribonucleides. This step has been replicated in laboratories.
3: By random chance, some of those ribonucleides happened to connect together with other ribonucleides into chains of ribonucleic acid, or RNA. Ribonucleic acid has a structure such that it can cause loose ribonucleides to come together into another, complementary strand of RNA (which can in turn cause the production of its own complementary strand, which would be a copy of the original), though on its own, this process is slow.
4: Strands of RNA also tend to get tangled into interesting shapes. Some of these interesting shapes would have the effect of speeding up the process of RNA duplication. Any strand of RNA which happened to come together in a way that would lead to those interesting shapes would tend to form copies more readily than other RNA sequences. Over a very long time, this will lead to those RNA sequences becoming more common in the chemical soup than others. This is the beginning of natural selection.
5: Once you have a variety of different RNA sequences which give themselves an advantage, sometimes you’ll have two shorter strands with different advantages come together to form a single longer strand, that gets both those advantages. These new strands are even better at competing against other strands, so over time, the RNA strands come to be dominated by things that have more and more advantages at replicating themselves.
6: Some of the advantages that these strands will develop include forming shapes that cause the amino acids around them to come together in interesting ways, too, including (among other things) ways that will make the RNA strand yet more efficient at replicating (these combinations of amino acids are proteins, including enzymes). Other advantages will involve some of the RNA strands being modified into strands of deoxyribonucleic acid, or DNA. Yet other advantages will involve the formation of a protective membrane to surround the other materials, so all of the interesting chemicals will stay in one vicinity.
7: Once you have all of these advantages (DNA, proteins, protective membrane), you have a living cell. This is such an effective combination for making a self-replicating thing, that soon, they outcompeted almost all of the non-cell replicating things, so almost all of the replicators were cells.
8: Natural selection didn’t stop once it had cells. There was still some variation between different cells, and whichever variants were more successful became more common. Sometimes a new variation would arise, and sometimes, that new variation would be more successful than its predecessors, and so that new variation would become common, too.
9: Ultimately, all of those random variations, some of which were more successful than others, led eventually (after billions of years) to all of the living things we have today.