What you need is fairly straightforward - the same things you need for any backpacking shelter. Wind block, precip deterrence, insulation, and anchor points. What's complicated - the sheer number of options a hammock affords you - far more than you have with a tent. It can be as simple as you want to make it, though.
The tarp is the equivalent of the tent fly - the size of the tarp is determined by whether you are a fair weather hiker looking to keep off showers or an adventurer who doesn't care if it's forecasted for wind and rain. I have a 9 x 11 caternary cut MacCat (four tie outs and two ridge line d rings) and a huge 11 x 13 tarp featuring four tie outs per side, which can be fastened down to the ground yet I can stand up inside (I'm 5' 7" tall). Most of the time I take the smaller tarp. I have on occasion closed up one end of the smaller tarp around the end of the hammock to block wind chill. It worked fine.
To combat chill from beneath, an under quilt or pads in the hammock - either works, and with a double layer hammock pads become less squirmy.
A very wide pad, a pad supplemented with narrow blue foam bumpers ala the Speer pad extender (these used to be up for sale til Speer hammocks went out of business, but are relatively simple to make) or with extra clothing around the edges, or two pads laid in a T configuration (shorter pad at the shoulders, to provide warmth for the torso) work okay. I have also in warmer weather used a hunter's blanket - the mylar-backed nylon tarp one sees sometimes in sports stores - and it does fine into the low 50s. It doesn't crinkle and is not so fragile as a simple mylar emergency blanket. Another version of this is the windshield solar reflector for a large truck - people score a couple of these at a discount store and take them hammocking. As on the ground, pads are additive. There are those who take pads into subzero temps while hammocking. Foam is perfectly comfy in a hammock unlike the ground... inflatable pads only need to be inflated to about half for hammock use, and are more comfortable in a hammock that way.
If there is much breeze, more like wind, something over the under quilt to cut the chill might be needed. Your poncho if you hike with one - or an emergency poncho, or a loosely hung trash bag, or one time I used my rain jacket along the windward side clipped to the edges of the quilt - you can be infinitely creative with multi use gear if you consider what you already have with you.
And the sleeping bag can be a quilt, if you find getting in the bag after getting into the hammock too clumsy. I find that the reduced bulk of an actual backpacking quilt is just right for hammock use. But plenty of people use bags in hammocks. Another trick - if you have a sit pad to use as a place to stand, you can put the sleeping bag on like clothing, standing on the pad, then sit in the hammock and swing up the feet. This helps if your hammock zipper (if you have a Blackbird, say) and the bag zipper are on opposing sides.
Using pads can add versatility. Here is my usual three season tarp set up in ground mode, in good weather at elevation where the trees became sparse. If there had been bugs I would have used the hammock as a bug bivy.
The hammock is the black tubular bag. It makes a dandy pillow when you are bivouacked on the ground for some reason. Alpine trees are too flexible to hold you up so I have done this twice - usually we are below tree line and fine.
Here is a Hennessy inside my large tarp with the far end closed up.
From the outside:
The thing about a hammock system is that you don't have to set up the hammock right away. You can put up the tarp, put down something dry to sit on, unpack the food, make dinner, have your friends in for cards, chat, and whenever you feel like it go about setting up - put on your rain gear again and get the suspension set, then string the hammock all up under the tarp. The rest of the pack contents come out when the hammock is ready to receive them.
There are other options, too. The pack and contents never have to touch the ground, in fact - it can be suspended from the hammock whipping. The hammock can be hung and then moved out of your way, by moving one of the stabilizing tie outs to the other side, pulling it against the side of the tarp. In one picture above I am repurposing a pack cover to hold gear and clothing, and it's dangling from the foot end of the hammock. I have also tied shoelaces together and dangled my shoes there, as well as threaded trekking pole straps around handles and hung them on the hammock. Marmots aren't going to eat my stuff again.
With a square of plastic or sit pad in the spot where your feet land, you can get in or out in stocking feet, or bare feet. In absence of a "ground sheet" I have put my crocs there. My dog, when she comes with, has the foam pad I use for sitting or standing. (This is a rare picture - a friend took it while I was still in the hammock. This is what it looks like "loaded")
Some of my personal data points as a reference:
Warbonnet Blackbird with whoopee slings and 6 foot tree straps: 28 oz
Two JRB 3 season quilts: 44 oz
MacCat tarp: 16 oz (with stakes and knotless guy lines)
5.5 lbs - the same as the old Kelty 2 person tent I used to have, but insulation included. the two quilts weigh about the same as the bag I started with. I used to have a shelter and insulation that weighed more than seven pounds, and then a Thermarest self inflator!
Lightheart Gear Solo: 28 oz
One JRB quilt: 22 oz
BA Q Core: 27 oz
77 oz, or 4.8 pounds - a whole ten ounces different. For less space, and less comfort. Hm.
Roll it again:
Lightheart Gear Solo: 28 oz
One JRB quilt: 22 oz
NeoAir medium (the original kind, not the new one): 13 oz.
Put my pack under my feet. 63 oz or 3.9 pounds.
The lightest option has the lowest R value. The Q core is more comfy and warmer. I have had both the Q Core and quilt AND two quilts on the hammock into the low 20s comfortably.
I could take probably 10 oz off the hammock setup by getting a half or 3/4 length under quilt and using the foam pad inside, under my feet.
"Hammocks are heavy" as compared to? well, yes. If you have a very light tent and a very light sleeping pad. Ten ounces is not going to kill me. My pack weight with the full hammock kit is still well below 30 lbs. including food and water, for 3-5 days.... a little closer to 30 if I throw in booze and fishing gear. As usual - priorities depend on your desired comfort levels, your planned activities, and your budget. I also have other hammock toys - a cover for the Blackbird's bug net for very cold weather hammocking, a DriDucks poncho reconfigured to be an under quilt weather shield, and a pad extender I made out of an old picnic blanket - but most of the time, the toys are left at home.
Hammock gear also tends to be exceptionally easy to make yourself, turning this into an easy way to backpack cheaply. Instructions on how to use a length of breathable nylon and a couple zip ties exist - add some Dynaglide or Amsteel line for suspension (you need something that has a very high break point, as physics of hammocks puts 700 - 1000 lbs of force on the suspension, so Home Depot poly rope won't cut it) and some tree straps, and get a blue poly tarp. Entry level cost minimal depending on how many of the ingredients you can scrounge.
The wider the hammock, the easier it is to lay flat. So a single person ENO is less comfortable than the double.
I made a Risk hammock (google this) out of strong silk. I haven't used it yet, but it's very comfortable and cost maybe thirty bucks. I need to add a ridge line and suspension. Playing with this stuff can be addictive.