In 2012, I watched Game 3 and 4 at LA Live and the area went crazy after game 3. Game 4 they lost so it wasn't crazy. Then for Game 6, they completely blocked off the LA live area so I didn't go down there. Who knows what they'll do this year but if you're able to get into the LA live area, there will be a ton of Kings fans watching at all of the bars there. They do not show the games on a big screen outside. You'll have to watch at a bar.
As far as where to stay, there are a few options. One you might consider is staying near Redondo/Hermosa/Manhattan beaches because if you end up watching the game in that area, there are a ton of places with lots of Kings fans. Then on top of that, it is a sweet area to hang out both the beach and the bar scene. If you're able to find a good deal on a hotel around LAX, you'll be close to the green line which you could use to get to DTLA (take green line to blue line) and about a $10 cab (or uber) ride to Manhattan or Hermosa piers. DTLA hotels will probably be pretty pricey but obviously really close to the action. It really depends what you're looking for but there are other communities near DTLA to stay at (West Hollywood, Santa Monica, Pasadena etc).
Also, I'd recommend using Uber to get around if you plan on drinking. Download the app and you'll save close to 50% off what a cab would cost and more convenient than transit. You don't have to worry about carrying cash or tipping. You load your card onto the app and it automatically charges your card and there is no expectation to tip. If you're traveling with a friend, get them to use your referral code when they sign up and get $25 for free when they do their first ride. You might download Lyft and Sidecar too. You can get promotional credit for signing up for each one and get free rides to save you some cash.