Seeing as I've set the bar rather high in DK with 240K, I figured I should provide some hints for any challengers.
A big part of my strategy in DK is to avoid jumping barrels, especially those nasty double barrels.
First off, I'll address the barrel bug, as it can really ruin a good run.
Barrel Bug:
Sometimes, when you finish the level, one of the barrels on the bottom level of the screen does not disappear. It continues to roll as Kong falls, and remains there when the next level starts. If it is in the same place where Mario spawns in, you're automatically dead. Fortunately, it does roll a
little bit while killing you, so it doesn't necessarily result in the loss of all remaining lives (though the same barrel can take two lives, or even all three sometimes).
To avoid the barrel bug turns out to be fairly simple. You just have to realize that when you finish climbing the final ladder, the bug might hit. So when you are doing that, keep an eye on the position of the barrels. The danger zone is the first few steps up from the flat part of the level. Any barrel already on the flat when you finish will roll past your spawn point if the barrel gets bugged. Also, I've never seen a barrel that is on a ladder stick around, so don't worry about anything coming down the broken ladder. If a barrel is just coming off the working ladder, it won't have time to roll all the way to your spawn point. If you get hit in this position, you have a chance to jump the barrel when you spawn in (sometimes you get the 100 pts, sometimes not). If you miss that jump, that barrel will likely take out at least one more life. I suppose you could try running to the left at spawn in, so that the barrel only kills you once, and is past the spawn point for the next life, but I've never tried that.
Avoiding the barrel bug basically boils down to keeping an eye on the barrels, and not ending the level when there is a barrel on the first few steps of level 1 (flat is fine, closer to the ladder gives chance to jump). Just get high enough on the last ladder to avoid barrels rolling underneath you, and end the level at the right time.
WARNING: It's not just the first barrel that you have to worry about. In my last game, I waited until the first barrel hit the flat, and the one behind it got bugged!! It hit me right on the spawn point and took two lives. So be careful of any barrels on the bottom level (I have never seen a barrel get bugged above that, and even if it did, you could just jump it).
--And yes, that means I posted 240K on one life. I got hit with the bug at 130K, lost my first two lives, and kept going to 240K before finally mistiming a double barrel jump.
Maybe when this score gets pipped, I'll shoot for 1 million
Basic Play:
At the start of the level, simply run and climb the only ladder to level two. Take the first ladder from there to level 3. Unless you had difficulty getting on the ladders, even the fastest barrel cannot reach you until here.
The "normal" barrel will roll to the end of the top level, and then come back and take one of the two usable ladders down to level 4. As long as you haven't had much trouble getting on the ladders, you can reach level 4 in time to jump the first barrel (it will be close if it takes the first of the two complete ladders down). If it takes the second one, you're fine.
After jumping the first barrel, place yourself between the two working ladders on level 4, and take whichever one the next barrel doesn't. Head for the ladder up to level 5, jumping a barrel or two along the way as needed.
The next bit requires some patience. Run past the ladder up to 5 and wait for either of two events to happen: a barrel takes the broken ladder down or Kong beats his chest after rolling a barrel. If a barrel takes the broken ladder, you've got it made. Leisurely climb up to the top and go save the princess.
However, if the barrels are all coming down to the end, you need to wait for a chest pounding. Then when the current barrel goes past, quickly get up the ladder and jump the next one (if you falter on getting started up the ladder, fall back and wait for the next opportunity). If the jump looks too close at the top of the ladder, you can also run to the right. If you stand on the far edge of the top platform, the barrel will go down the ladder and just miss you.
That's the "normal" script for a level, but of course, barrels do not all take the same paths. You need to watch out for the deadly "double barrels". These can form when one barrel takes the long route down, but the next one comes down the broken ladder. Note that double barrels can be jumped, but your timing has to be very good. It's much safer in the long run to avoid them.
If the second barrel of a level comes down the broken ladder, I usually will stop at level 3, and wait just past the middle ladder. If both barrels pass by and go to the far ladder, I can climb up the middle. If they split, I can either wait for the one on the middle ladder to pass, then climb up before the other one gets to me, or I can jump the other one and take the far ladder myself. If they both take the middle ladder, I just go climb the far one.
If you are already on level 4 when a double barrel forms, the best place to be is in between the two working ladders. If both barrels take one, you can take the other. If they split, you want to stand as far right as you can while avoiding the one on that ladder, and as the other reaches the bottom, quickly jump it. If it was the first barrel that took the ladder on the right, you can also try climbing up immediately after it passes, but if you stuck trying to get on the ladder, the other barrel will get you.
If you're on level 4, but can't get between the two up ladders when a double barrel is coming, you can try backing up past the middle ladder going down. Hopefully, at least one of the barrels goes down that one. If they both keep coming, you'll need to get your timing down to jump them both. Do not attempt to hide on the far right side of level 4, you're not safe there like on 6.
Once in a while, you'll get two barrels that are not right next to each other, but close enough that there is little room to land and jump again between them. In this case, you should try to jump the first one with a stationary jump (jump straight up), and then jump the second normally. These can only happen on level 4 or lower, so you won't have to deal with them often (optimally, you position yourself on level 4 between the ladders before you encounter the second barrel).