Garlic can be grown hydroponically, but as anyone whose tried will tell you; it just smells rank.
Also, garlic grows 18"+ high which gets the tips of the plant up there near the light, but leaves the lower parts poorly exposed to light. For this reason, either grow outside, or use vertical lighting for garlic.
Buy a clove at a grocery store. Use the best you can find, because as they say, "Garbage in, garbage out!" Separate as many cloves as you want planted. Don't peel them. Plant them into coco coir, perlite, or vermiculite with the fat end down and the narrow end up, just below the surface. Dampen them a bit. If you let them grow to maturity, new cloves with generate and divide, and for this reason, do not use a mesh basket that is too small.
See this article on how to start garlic. They simply use garlic in a glass of water, and they recommend keeping the lower half of the clove submerged in water to start the plant.
System Type Edit
In nature, the garlic cloves are very near the surface, and the roots grow downward. The clove is not meant to stay damp all of the time, and so will rot if this is the case. An NFT system with only the roots in the film stream is suitable. An ebb and flow is not.
Garlic is a full-sun plant, so HID lighting is pretty much necessary.