What Makes Him Cheap: His 3rd ability, the passive. This is the only real ability to counter b/c even duel is not all that powerful. It is like a stun, but only works if you are 1v1 or fighting weak heroes. So to counter LC, you need to counter this 3rd ability. The ability is similar to Axe's passive helix. The more he is attacked, the higher the chance he'll get an increase IAS with massive lifesteal. He can be redline and instantly regenerate all his hitpoint.
Hero Counters: LC is vulnerable to stunners and casters. This is also the reason LC will duel stunners and casters. This means you need at least 2 or 3. This doesn't mean you need a bunch of INT heroes, there are STR stunners too. But not just stunners, LC is vulnerable to nukes. Remember, we're only required to counter that 3rd passive ability. Other protector heroes can also counter him such as Roof, Oracle, Phoenix, etc... Range DPS is also good, but only when he's not attacking anyone b/c otherwise you're just helping his passive.
Item Counters: If you are a support hero and you're afraid of being picked off, you need a ghost scepter or euls. Don't be cheap about spending money on defense. I know many people only rush key items. But the point of the game is to stay alive and deny your enemy food. Before anything though, most LC go lothar. You must buy early dust and wards, don't be cheap. It will be worth it. Preventing him from getting easy pick offs will keep his level and damage in check.
However, the best counter item early is actually a dagon. Dagon is one of the most misunderstood items in the game. People think caster items are only for casters. Items are either chosen to enhance your heroes abilities OR supplement their weakness. When I say dagon counter, I mean dagon on your non-supporting heroes. If you have 3 support heroes with nukes and stuns, LC will be targetting one of them for duel. He won't be expecting your other 2 heroes to have 400 damage dagon. When LC goes redline, you want to finish him off with that dagon or he'll regain his hp fast.
Now, of course, not every hero can benefit from a dagon. But consider this, many STR and AGI heroes can use a larger mana pool, right? It actually makes less sense for INT heroes to get dagon when they already have larger mana pools and they're going to get much higher level INT items. Dagon is meant for non-INT heroes to supplement their mana pool and giving them a 400 damage nuke.
Another reason is dagon's surprisingly cheap price. For its abilities, it is extremely affordable and can be easily bought by level 6 in a dual lane. Think about all kinds of items you can purchase by level 6, and how much each increases your hero's effectiveness. You can get a dagon by level 6 without having much of a decrease in your farm timing on your hero's key items. In fact, used properly, dagon can help you farm and make up for its own cost by killing heroes.
These are early item counters to keep LC in check, so he doesn't grow. It will force him to go BKB, which again is far better than him getting his key items b/c LC is item dependent and needs items to be effective. He may also go BM if the range DPS are giving trouble or even the nukes. Again, far better than him getting other items b/c BM is not really useful if you do not have the HP to tank. BMs are easily counterable with euls and ghost, just delay until it turns off.
Late game counters will involve guinsoo, etherals, etc... b/c even if he uses BKB/BM, it has a time limit. If he doesn't get any kills in that time and the BKB wears off, he's done. For carry heroes, melee should definitely get basher and IAS. Linken can counter duel, but it isn't worth it in my opinion b/c it can be triggered by any point spells and seems awfully expensive just to counter duel when duel is not even that powerful. That's about it. He's not really that hard to counter.
Strategy Counter: Once the lane are down, your team needs to stick and watch out for each other. Duel is useless in most team fights and when you have enough supporting each other. Remember to dust and ward early for Lothar LCs, deny him easy picks.
When he duels an ally, you want to disable him of course. If he BKBs, it is actually better NOT to attack him until the duel is finished or BKB wears off. Remember, each time you attack, you increase his chance of getting that IAS and lifesteal. Your teammate getting dueled has a higher chance of survival if you didn't intervene in some cases. If you are caught in a team fight and he duels someone, focus on taking out his allies such as supporters or weak range heroes. So even if he wins the duel, he'll be left alone and you can finish him off. Many people don't seem to be expecting this. I do it all the time, I see a teammate getting dueled and I can't disable, so I look for their Rhasta or Sniper and take easy kills. Remember, duel or not... you don't focus on the tank, always take out the easy kills.
Summary: This is pretty much it. He's not that hard to counter. I think most of the time his "reputation" is more scary than the actual hero. Most people just get scared when they see LC b/c they don't want to get dueled. This coupled with most people simply don't bother getting correct counter items or even bother buying wards and dusts. Spend the money on the right items, keep LC down and keep him in check, deny him early pick offs and he is just like any other hero.
Now go out there and own some LCs. Tired of seeing this hero around.
You're teaching people how to play LC!!! Stop it!! 
But I originally wasn't planning on a dagon b/c they have some heavy lategamers and I wanted to go lategame counter against them. However, early in the game I realized a few things.