Lich mid and Lich offlane are played pretty much the same. The perk to being in either scenario is that you get bumped from being a 4/5 position (support and hard support) to a 3 (semi-support-carry-master-of-death). Your prime objective at that point is to disrupt their mid and setup ganks. How do I do this oh wise master of all things Lich? Simply put, you be an asshole. You block the creep wave as hard as you can and sacrifice creeps from behind the tower. Sacrificing creeps give you the exp value of the creep, and denies your opponent that (as long as they aren't in range).
Get yourself a quick pair of boots and ward the rune spots and steal or deny every rune you can. Remember, your goal isn't necessarily to win the lane, just to minimize what your opponents accomplish. When you hit 6 or 7 (depending on how the game is going, and you should have a skill layout of 4/0/2/1 prioritizing ult>frost blast>sacrifice) lurk towards whichever lane they are pushing more and wrap around them while pinging like a sociopath so your teamates are fully aware the Lich Train is coming.
Now, when engaging, Lich requires a bit of on the fly thinking. Don't just Chain Frost the first hero you see, make sure if you use it you're guaranteed a kill. If you can get the jump on 2-3 heroes persuing your hard carry odds are you're going to kill them both. If it's a group of 3 always focus on the least mobile heroes present, something like a Lion, Shadow Shaman, Lina, WD, etc. If you can get a Chain Frost off between 2 slow supports, they have nowhere to run and are most certainly dead.
As a little aside, I usually don't go past 2 or 3 points in Sacrifice until later. Frost Armor is really good in the mid-game and can really screw people up. As for items, normally I play full support Lich and go for something like a Mech, but if you're in the 3ish position you probably want to go for some wonky utility items. If your farm is incredible get a fast Scythe, eventually you will want one but no reason to waste time with only a Mystic Staff when you could get cheaper better items earlier. While typing this although I've never used it I feel like Rod of Atos on Lich would be nice if you have a gank heavy team, and maybe even something like a Eul's would be good. Agh's is nice for fights if your opponent is very melee heavy and lacks mobility, but in most cases it won't help a ton.
Hope any of this helps.
