[RPG] What do I need to play Lamentations of the Flame Princess

Replacing analogous monsters with their equivalent from your actual system (D&D Next in this case) is the right way to convert encounters in OSR adventures, but for when there isn't an equivalent, yeah, it's nice to have a quick conversion method.

Armour Class

AC is pretty easy: LotFP AC starts at 12 and goes up, D&D Next AC starts at 10 and goes up, and 1 point of AC is the same in both. To convert, drop 2 off LotFP ACs to get D&D Next AC.

For humans (and dwarves and elves, etc.), don't do a direct AC conversion: instead, recalculate their AC according to what armour they're wearing.

Hit Points

LotFP monsters use a d8 for Hit Dice, giving them an average of 4.5 HP per HD. D&D Next uses variable HD sizes, but they seem to vary from d6 to d12, so we can consider either a d8 or a d10 as "average" monster HD size in D&D Next. If you go with d8 then no changes are needed; if you go with d10 then that has an average of 5.5 HP per HD, 1 HP per HD more than LotFP.

OSR adventures often, but not always, have the HP pre-rolled; in this case, just add 1 HP per HD to get an HP total closer to D&D Next's system expects. If the HP aren't written, just roll some d10s when the encounter begins or (to use an old DMing rule of thumb of using the rounded average) arbitrarily say HP is 6 × HD.

Rolling is slower, but has the advantage of giving you an easy way to inject some character into the encounter – describing one orc as especially big to reflect high HP, or an ogre as especially sickly (or perhaps already injured) to reflect low HP.

The only trouble is that D&D Next monsters tend to have a lot more HD than their OSR equivalents, resulting in lower HP totals. This isn't something you can easily convert, since the D&D Next team seems to be (judging by the quantity, and the way they're using different HD sizes) making these stats up anew instead of basing them on anything that went before in a formulaic way we can reverse-engineer to a LotFP equivalent.

Eyeballing it, I'd say monsters tend to have half again to twice as many HD as they did in old-school editions. You could just give every monster 1.5× or 2× its HD. Alternatively, once you've read some of the "non stat differences" I talk about below that will make the adventure harder than it seems on paper, you might just leave the HD totals low to make it somewhat easier again.

Speed

A normal human's speed is noted as 120' in LotFP. D&D Next, a human's speed is noted as 30 feet, which is exactly 1/4. Divide all LotFP monster speeds by 4 to find how far they can move in a single move action in D&D Next.

To-Hit

Monster in LotFP have a to-hit bonus that is equal to their HD. D&D Next has a much different system, which lists a specific bonus with each attack a monster has, and the bonus tends to be a little or much less than their HD (for their main attack) or less, sometimes down to no bonus at all. This is the one place where I've no idea how to adapt them properly. If I was doing this on the fly I might give a monster's primary attack a to hit bonus equal to the original (unadjusted) HD, or half that if it's a "caster" type of creature.

Damage

Damage is pretty much on par, and doesn't appear to need any adjustment.

Saving throws

LotFP creatures use the saving throws of a Fighter of the same level as their HD. You can just use that, instead of trying to translate the numbers with math, since that will give more consistent results than trying to math it. Though LotFP doesn't do this, you might consider giving "caster" type monsters saving throws according to a Cleric or Wizard, but really you can just leave them as Fighter type saves without difficulty.

XP

This is the hardest conversion. Most OSR monsters don't include XP in their adventure entries, because those are usually in the game's bestiary, or not listed anywhere and instead calculated based on HD. They're usually calculated as a certain amount of XP per HD, with monsters with "special abilities" counting as having bonus HD for XP purposes. Unfortunately, XP is very system-specific, and D&D Next's playtest hasn't revealed its "how to build monsters" math, so you're kind of on your own here.

Important non-statistical differences

There are a few major differences between "modern" D&Ds and old-school D&Ds that can trip up a DM using an OSR adventure in a modern D&D. All of these have an influence on the original numbers in an OSR adventure, but in ways that are very hard to work out a formula for conversion to a system that doesn't have these features.

Hirelings

That the PCs have followers is expected in the OSR, and adventures are written with the expectation that there will be about 2 to 3 times as many mortal souls entering as there are PCs. (And, that some of those extras won't return.) There's no easy forumla for how to scale the adventure down if your PCs don't have followers, so this is something you have to eyeball. Fights might have one or two monsters left out; single monsters might have fewer HP; traps might be less deadly (whatever that means); but equally, changes may not be necessary.

Since more or fewer bodies creates a non-linear difference that can't be easily translated to equivalent stats, there's no easy way to quantify the benefit of having NPC followers in a way that compares against monster stats. This is something you'll have to keep in mind when reading the adventure, and consider whether the situations presented in the adventure might need adjustments.

Not every monster is an enemy

A hallmark of the OSR is that even the DM often doesn't know how the adventure is going to go, partly because they don't know what the monsters are going to do until the PCs start interacting with them. LotFP has a Reaction Roll rule, where you determine whether the monsters are hostile, unfriendly, neutral, indifferent, or even friendly. (All pre-3e D&Ds have a similar reaction system.)

Consequently, not all monster encounters will result in a fight, since a reaction roll of anything but "hostile" may result in a parley, the monsters retreating before the PCs even notice them, the monsters telling the PCs to go away, a chance to hire the monsters to help the PCs, or some other variation of "not a fight".

Since not every encounter in an OSR adventure is expected to be a fight, there tend to be more of them than in adventures built for modern D&D adventures. Again, this isn't as easy to quantify as simply counting the number of encounters, since the reaction roll table (and its unquantifiable possibilities of intimidations, ambushes, fetching allies, forging alliances, etc.) has an unpredictable effect on the social-roleplaying level of the game. Again, this is something to keep in mind when reading the adventure: "encounter" doesn't equal "fight".

Not everyone fights to the death

Morale is a very important rule in pre-3e D&D, since it means that fights are often won by bloodying the enemy, not by obliterating them. Morale rules vary considerably among D&Ds and retroclones, but generally it's rolled the first time a member of a group falls and again when the group is reduced to 1/2 its numbers, or (for solo monsters) when a monster is reduced to half HP. The result is usually fleeing or surrendering, depending on what their situational options are.

As a result, there are yet more encounters in OSR adventures, but encounters aren't always fought to the death. On the other hand, this makes undead and other high-morale or never-checks-morale creatures especially dangerous in OSR adventures, in a way they're not in modern D&D.

Random encounters happen

Random encounters are a feature of every pre-3e D&D edition. They are not always written as part of an adventure, since there are already rules in the game for generating random encounters, but they're expected when someone designs an adventure. They provide part of the choice/consequence dynamic inherent in old-school style adventures: do we spend time exploring X (at the risk of something finding us), or do we try to be quick about things? Random encounters are, like all encounters, not the same as a fight, but they provide a pressure that factors into how most OSR adventures are written.

But you can ignore all this

Despite all these confounding factors, as a DM running an OSR adventure you have a safety net that means you don't have to worry about balancing the adventure: OSR adventures are not railroads, and inherently expect the players to exercise their free will about when, where, and whether they get into fights. You can adjust only the stats and leave everything else as-written, and run the adventure just fine, if the players know they can (and should) exercise their good sense in whether to press on or to retreat, possibly all the way back to town.

This freedom means that, if the players find part of the adventure particular hard because you haven't massaged encounter sizes and frequency to try to account for the differences that hirelings, random encounters, morale, and reaction rolls have in a true OSR system, they can simply avoid that part of the adventure for now. They might just avoid it completely, or they might decide to return to it later and get really excited about planning how to "beat" it creatively.

This is a good outcome, and is also an expected part of OSR play that influences how adventures are written: in an OSR game, not every encounter gets fought, not every fight is won, not every secret is discovered, and not every treasure is found. You can use that to your advantage, letting your players run into hard fights, easy fights, and generally let them decide how to do the adventure according to how hard they find it. This means you don't need to carefully adjust everything to be perfectly beatable, which saves you the trouble of trying to do something that is nearly impossible to accomplish anyway.

You also need to discuss expectations if you haven't already

If you just change the mechanics to better fit the style of play that a highly lethal old-school module is written for, you need to discuss the expectations of this style of play with your players too. If you don't talk about these expectations, all that your players will see is that you've changed the system and made them less powerful apparently for no reason. I would highly recommend having your players read a quick primer on old school play; if you can get past the bias inherent in the document, it's very helpful. Even if you don't have your players read the primer, you can still state the main points and present the reasoning behind you've changed the system. Making sure players know that build is less important, that death is frequent due to lack of caution, and that player skill is rewarded is going to be necessary to support the style of play you want here. Other than this...

Running an LotFP module with 5e is perfectly viable with some conversion work

As long as you convert the module's mechanics over, running Tower of the Stargazer should work fine. The main issues would be hit point inflation (which is sounds like you've covered); the Perception and Investigation skills (which it also sounds like you've removed), and Saving Throw DCs. I'm assuming (based on your question) that you feel comfortable making mechanical changes to both the system and the module, so I don't think this will be a problem for you. One point of caution I'd like to advise on, however, is length of time spent to make a character.

Avoid Lengthy Character Generation

The general expectation of a game like LoTFP is that characters die frequently, and thus, must be created frequently. You'll want to keep the time a player spends out of play due to a dead character as low as possible. A LoTFP character can be created somewhat more quickly than a 5E character, so I'd advise making some of your system changes reflect this. Particularly; I'd say that you should have a list of cantrips and 1st level spells available as handouts for your spellcasting characters so selection goes more quickly, that you either enforce rolled stats or use a standard array for scores to not spend a lot of time on point buy, and that you don't use feats and skills (which it sounds like you've already done). This is just a very quick and dirty method of reducing time it takes to create a new 5e character, but hopefully it will work for you.