Cells are thought to have first appeared about 750,000,000 years after the formation of the earth about 3,800,000,000 years ago. These theories are all speculation as the conditions are obviously not recreatable in a lab. One theory is that the first step in forming cells was the spontaneous formation of organic molecules. While now this seems silly, back then the atmosphere was mostly CO2 and N2 with small amounts of other stuff including sulphur, phosphorous. We can and have proved in labs that given an energy input such as sunlight or electricity. Organic molecules, such as amino acids will just form spontaneously from environmental constituents, [this was shown by Stanley Miller in 1953}(http://archive.org/stream/nasa_techdoc_20050139759/20050139759_djvu.txt).
So you have the basic building blocks, how do they make other stuff? again, spontaneously in those conditions, they can form together chemically at random forming bigger and bigger molecules in conditions seen in early earth. These eventually formed RNA and even longer, its big brother DNA which make up, well, you.
The other thing you need is the membrane and this is a tad simple. A membrane is made from something called phospholipids, a molecule, which has a end that loves water (hydrophilic) and an end that hates water (hydrophobic). So basically one end wants to be in water, and one doesn't. Imagine oil on water! It forms bubbles same with these molecules, they group to gether and form bubbles with the hydrophobic ends out of the water in the center of the membrane and the hydrophilic ends not. This then forms a crude membrane called a phospholipid bilayer.
So you now have your RNA/DNA and your membrane. From here the RNA can self replicate, and change forming other proteins such such as more molecules for the membrane and some proteins that eventually lead to life over billions of years.
