Fishing with lines through the ice is not a particulary effective means of catching fish. But as with traps, they take little effort to set and they can fish for you while you are not present.
To achive a reasonably steady supply of trout throughout the winter, you are likely to need at least 20 lines out. Some days you will have many, but you will at least be almost certain to catch one every day. This of course depends on how numerous the fish is in the area in question. In some places you may get by very well on 20 hooks, other places that is way to little. If you have to set more than 20 hooks to get fish every day, I would seriously consider moving to a better fishing location. It takes time to check all of those hooks and if the return on them is poor, you may be better off spending more time on setting traps. But as a general rule: Fish is a more dependable food source than game.
One of the main advantages of fishing through the ice is that you need comparatively less line than when fishing from the bank. This being of the simple reason that the ice helps you drop the line straight down on the fish.
Setting a line and hook:
First, of course, make a hole through the ice. Make it bigger than you expect the fish in the lake to be. Scoop out the ice bits and snow out with your hands.
Bait the hook with something you know the fish like. Trout likes worms and other small creepy things. But these are hard to come by in winter, so they have to be stored. The Saami use reindeer fat. I have tried elk/moose fat and have yet to have any success with this. Other, more aggressive fish like pike, seems to like pieces or whole fish. Old bait doesn't work very well. The bait should preferably be changed every day.
The fish usually stand just a little over the bottom of the lake. So that's where your bait should be. Use a sinker to get the hook down. I often find that the best fishing locations to be where there is little water under the ice, maybe just a metre or two. Especially in the spring. In mid-winter they often stand deeper. Below: A line with a baited hook.
Tie it or wrap it securely around a stick at the desired depth. A few motions on the hook can often give you a fish right away.
Don't try to lift big fish directly out of the hole. Everything weighs less in the water. Grab it by the gills and lift it up. I have saved a lot of fish unhooking in the hole, by grabbing the confused, but freed fish down in the hole.
Make sure the line doesn't rest on one of the sides. Otherwise, when the hole freezes over you will have a lot more chopping on your hands, with the risk of cutting the string of course. Cover the hole with snow to reduce the freezing. Spruce boughs can be laid underneath, but I usually just showel a pile of snow over the whole thing. I have seen the Inuits make a small igloo over the hole, but I don't find it to give any advantages over the previous methods. If anything, it has to be less snow in the hole.
About safety. Check the ice on intervals to avoid falling through. Especially where there is fast flowing water. Carry a long stick if you are insecure about the ice. Then you can use it to climb back up on if your luck turns sour.