This is how I run Skills in LotFP. It's also rather similar to how they work in Wolfpacks. Odd, that.
So, first up, if you have a 6-in-6 chance on a skill roll, you roll two dice, and if both come up a six, you fail. Otherwise, you take the better of the two rolls.
Likewise, if you have a 0-in-6 chance, you roll two dice and if both come up a 1, you pass. Otherwise, take the worse of the two rolls.
Skills can't go above 6-in-6 or below 0-in-6.
By and large, you won't be rolling for skills at all. I'll judge how stupid or smart the plan is, and you just succeed or fail based off that. Sometimes, I'll just look at how good you are at the skill, with no roll; for example anybody with at least 2-in-6 in climbing can probably get up a cliff face just fine.
I try not to use roll-under-attribute rolls, since skill rolls have way less chance to succeed than attribute rolls. By and large, I'll use the closest approximate skill to whatever you're doing for a roll.
Your attribute modifiers apply to your skill chances - that's bonuses AND penalties.
Strength applies to your Climbing and Open Doors chance.
Dexterity applies to your Stealth, Sleight of Hand and Sneak Attack chance.
Intelligence applies to your Languages and Tinkering chance.
Wisdom applies to your Architecture, Bushcraft and Search chance.
This bonus applies to the basic 1-in-6 chance everybody gets, to the improved chance a specialist gets if they've put points into a skill, and to a non-human's improved skills. You might occasionally get told to use a different modifier for the skill - for example, modifying your Stealth by Charisma rather than Dexterity to blend into a crowd without attracting attention.
There's four unlabelled dice for skills on the base LotFP sheet, and I'm never going to let the opertunity for homebrewing go to waste. As such, I've come up with four new skills to fill these gaps, detailed below. On top of this, I've altered how Sneak Attack works to fit in with my damage system.
This skill lets you know what that treasure you got is. A successful Appraise roll lets you tell if an item is magical (and therefore chaotic) or holy (and therefore lawful), and perhaps gives you a rough idea what it does. You might know that a sword is magical, or that a cloak is protective, but not much more than that. Appraise rolls also tell you how much the treasure is worth to a collector, to a pretty accurate degree, and might let you notice other useful features. It uses your Wisdom modifier.
Charm lets you deal with NPCs better. You can use it to slip a lie past somebody, to fast-talk or intimidate somebody into doing what you want, or to get your hirelings to do what you told them to when the going gets rough. A successful Charm roll lets you modify the result on the encounter reaction chart or a hireling's morale or loyalty check by the amount shown on the dice.
Charm is modified by Charisma. I mostly put it in so that all the rolls you might need to make can be skill rolls rather than attribute rolls, and Charisma didn't have a go-to skill.
Medicine lets you fix people when they get broken. What it does is basically summed up in my post on hit-points, damage and healing. As I state there, it uses your Intelligence modifier.
Research lets you find things out in down time. It doesn't matter if you're going through books in an old library, listening to gossip in the pubs, or interrogating prisoners, it basically works the same. If there's doubt over if you'll find the information you need, or if you'll find it in time, then make a roll. If you can just keep on going, eventually you'll find what you're looking for or somebody will blurt out what you want. This way, if you just say 'I keep researching until I find out' you can (slowly) learn what you need, and the game doesn't stall while you roll dice to be allowed to continue. Research uses your Intelligence modifier.
Rather than multiplying the damage you do like in basic LotFP, this works a little differently. When you want to shank somebody, you'll need to be in a position to do so properly. Snipe at them from concealment, get behind them, attack them before they know you're armed, whatever. So long as you catch them at a disadvantage, you get to make the roll. If you succeed on the skill roll, your attack (if it hits) deals its damage straight to their flesh.