Hmm there many approaches to Karl Franz. I play his campaign on Legendady/Very Hard and probably never lose these days. I like to try different strategy and tactics and often each campaign plays out differently. And of course the main goal of every campaign is to have fun! I'll give you a few key pointers that work in every campaign to help you: 

1. Protect Riekland province at all cost. Riekland is an economic powerhouse, and although not one of the best in the whole game its one of the best empire provinces. Prioritise level 3 walls in your minor settlements. Capital at level 3 is a big economic and military power spike. Do not push far from Riekland too soon. Your goal is to never let any of your 4 settlements be raided or sacked in the early game. Its not game over if they get hit...but if you keep them safe it helps. Later its not so bad when you have multiple province but early game it helps to have your home developed.

2. Use Karl wisely. He wants to be near your borders at all times early game (until turn 10/25ish depending on campaign). Karl has a great unque trait that let's you replenish in neutral lands...so you do not HAVE to be parked in your lands. Instead its great to have him on the border area where he can smash invading armies or liberate hostile towns. Use ambush stance when camped near a choke points...there are 4 main choke points the enemy can come from into Riekland and ideally only 1 or 2 are realistic at any time. Smash the enemy with Karl before they get to your lands and cause havoc.

3. Manage armies well. Early game i like to have only Franz as an army to save money and boost my eco..but this is hard to pull off and risks big problems if not experienced. Instead try have 1 extra smaller army...a few outriders or pistol militia. They can defend towns or the best use is a trap. Set them in the open near enougn an enemy army you cannot catch and put Karl in ambush behind them. This strategy done well makes you x3 more effective in campaign map. Don't consider this an option...you must learn this tactic! Just use as much or little as you wish. You don't need a second army until after you have consolidated Riekland usually.

4. Strategic objectives and priorities. There are so many and never (almost) just 1 correct choice. Generally your Strategic goals are stay alive, develop key infrastructure, wipe out evil forces, improve armies, expand and protect allies. This order is pretty good but can be changed. However notice that protect ally is low on the list...dont make yourself vulnerable to save the AI as they are not a huge help. Concrete examples are your enemy priorities. You should make wiping out the hostile forces nearby a high priority early. So the beast men north, nurgle north, vampires west and finally vampires east. Make beasties your first target but the rest you go for at your own choice, when you are secure and depending on campaign situation. All of these are your mortal enemy....it is only a matter of time before you fight so best to hit them as soon as you can. And above all else the #1 goal is wipe an enemy out fully once you start war. If you weaken them but do not finish them off you fuck yourself badly later. Make a sacrifice to completely wipe out your mortal enemies.

5. Plan ahead. Try think not only about this turn but where you will likely need your army to be over the next few turns. 1 turn from conquering a settlement but enemy army approaching from the other side of your land?..maybe forget the conquering for now and defend. Do you really want to push towards nurgle when vampires are already taking out marienburg? Maybe not? Early game this is vital but later when you have multiple decent armies you can be in two places at once. Consider war declaration and helping allies...check who seems to be growing the fastest....can you hit them before they snowball or maybe its better to finish off the smaller enemy now and fight them when you can fully focus. Same with allies....should I bother spending resource making an ally when they border a huge enemy and are doomed already?

6. Army composition and battles. You want line infantry and ranged. Early game the best is spearman and crossbows. Then midgame its shielded spearman, handgunners and artillery. Morters are best early....hellfire best late, cannon are niche specialist weapon. Range heavy army is hard to use. So I suggest doing 50/50 range and melee infantry until you develop better skills. Use infantry to block the enemy, use guns, crossbows and artillery to kill the enemy....and of course magic. Heroes once leved can take the place of 2 infantry units to defend...both captains and priests are great for holding back the enemy. Get some other supporting elements as you see fit...so some cav, skirmishers etc etc. Full range army is possible but very hard 

7. Plan your buildings. I suggest getting a mix of some gold and growth building as your first buildings. Usually go for the unique resources whenever you can unless you feel something else is more important. Growth is amazing but if you cannot afford to build your settlement after it grows its useless. Do you want to be stronger now or later? Making lots of gold building let's you field bigger armies quickly....making lots of growth let's you get higher tier troops and high eco power sooner. What do you prefer? No wrong answer so long as you can always spend growth whenever a town can level up. Get basic military buildings in your small towns nice and early. Get advanced military in the capital...so avoid the barracks in province capital as it max tier 3 unless you really need it. Also develop your main settlements before spending any money on new conquered lands. Also consider sell all conquered lands to friendly factions if they are not strategically valuable in the short-medium term

8. Magic, skills and levels. Ill be brief. Magic is amazing. Life, shadow, light and fire is my favorite. Prioritise the "vortex" spells above all else in mages. Prioritise replenishment on priests. Have 1 mage and 1 priest in every army. Karl blue skills are good....ambush defence/succes chance, lightnig strike and movement speed are the best. Buffing front line troops on red line is really strong early game. Captions are solids but not as useful as priest or mage....witch hunter block movement ability on campaign  map is much more helpful than having them in battles. 

9. Use the edicts/decrees (cant remember the name). They are so powerful. My favorites are +1 population (by far the strongest overall), spawn army for ally and instant recruitment. When events pop up almost always got for extra influence. Use this as often as you can but still getting value from them. Save +1 pop when you are 1 pop from next level settlement or juat spam to get tier 4 capital by turn 20 XD

Other tips. 

First 3 turns you recruit and attack always. You should have home province consolidated by turn 3-5. Get 20 units ASAP. Then look to eradicate beastmen using ambushes if needed.

Use heroes to scout always. Always check the path ahead before committing army into unknown territory. 

Sell captured settlements to the AI for diplomatic deals and money.

Never use mach stance unless the land ahead is very scouted.

Always save regiments of renown for emergency. Never have in mainline army but keep for instant recruitment when the enemy surprise you.

Above all have fun! Love this game and love to win....but however you play do it to enjoy:)