Speaking from beta experience (I don't think the KV play style has changed all that much):
The KV requires you to be flexible between two vastly different styles of play.
Style 1 - Low Tier (5-6)
In a Tier 5 game, you're basically king. Many lower tier tanks will bounce shots off your frontal armor all day, especially if they're driven by newbies who don't know how to aim. There was one game in beta that my KV survived after tanking something like 80 hits*. Many enemies can be killed in 1-2 shots. Seriously, the 107 is just sick at this level. Bear in mind that this gun is only slightly worse (20 less damage, 30mm less pen) than the gun US heavies get at Tier 7 and are stuck with until Tier 9. At Tier 5, you really just have to be careful not to over extend and let yourself get flanked. Tier 6 is a bit more dangerous, as M6's, KV-3s and VK3601's can all lay a hurt on you, but you still dominate the lower opponents.
Style 2 - High Tier (7+)
Here the tables are turned. Basically everyone can penetrate you, and most opponents will be able to kill you in 1-2 shots. Most of the time, once they start shooting at you, you're dead. Your entire mission is to sit in the back and snipe. If you can avoid being a target, you can actually chalk up quite a bit of damage on higher tier enemies thanks to that beautiful 107, but you're going to have to learn how to shoot manually and where various tanks are weak. Tier 7 enemies aren't too bad, but once armor values start shooting up at Tier 8, you'll start having a harder time getting penetrating shots.
*This was on Cliff. I was at one end of the valley, and they had 3-4 lower tier TDs that sat at the other end plinking at me the whole game without penetrating, while their heavies and meds were attacking.