I am a heavy coffee user and I want to know what's in my coffee and therefore I'd like to grow coffee in my home.
Is it possible to grow coffee in an apartment?
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It is possible to grow coffee indoors. If allowed to grow as a tree they can reach 8 feet tall but if trimmed occasionally you can get the plant to grow more like a bush around 3-4 feet. This will change some based on the exact type.
Coffee plants prefer shade with a couple hours of direct sunlight in the morning. Keeping it by a window should be about right. Depending on your climate you want to be careful taking it outside during winter.
As J Musser mentioned, this will be a long process that will yield very little, but it can be done.
On the plus side, coffee plants are evergreens and look very nice as indoor plants.
You can read more here.
Edit: Stealing @JMussers comments to add to my answer
One large apartment plant will only produce enough coffee for a few cups worth each year, and that's after years of growth. They do look good, though. My point was, indoor coffee plants aren't that effective. Outdoors, a good plant will produce around 2-3 lbs. of grean beans per year. They are usually planted about 8'x8' apart, or a near equivalent, so 680 plants per acre, = around 1700 lbs of green coffee per acre per year, under good conditions.
If you have enough space in a garden or you can rent a piece of community garden, you can consider setting up a greenhouse. As such, you can control the optimal environment for your plants and don't need to invade your living area.
In an enlightened environment, but not directly under the sun, keep the temperature in between 15 °C and 25 °C, use plenty of water to the soil, water once a week when hot, water once in two weeks when cold, don't eat its leaves.
As the title itself asks for how to grow, I would like to answer that also.
A few days ago, I have encountered an Arabica plant. A very small one, at most one years old. At the bottom of the pot there were these instructions illustrated. I place the photo of the tag below. Happy farming!
Note that, Cenaphora (aka Robusta) or Liberica may have different instructions. Meanwhile, as Liberica is a very large tree, I don't expect anyone tries to grow them at home.