The computer is quite poor at defending against attrition tactics, and these can therefore be used to defeat him. This is even the case if your opponent is much stronger than you are.

The perfect attrition tactic is one where you inflict more damage than the losses you take, however that isn't always the case, and simply being able to replace your losses faster than your opponent can be good enough.

1. Attack, Attack, Attack.
The first step is always attack, everywhere all the time. Attacking doesn't actually do much damage, but it reduces organisation and prevents the defending troops from moving. The computer constantly moves their troops around, and of course, looks to attack you. By attacking him, you are stopping him from doing so.
2. We need more men... or do we?
The best forces for attrition are infantry. Lots of infantry. Due to the nature of the game however, it is unlikely you are going to be building lots of infantry. If you are going to use attrition, it is going to be because you *have* a lot of infantry. If you have a high industrial base, you will be able to build better forces and not need so much infantry. What you need to do is make your infantry the best possible. This means spending peace years researching the latest plans, and upgrading them all. Upgrading your infantry is far more important than building more. There is a bit of a trick to this if you can do it right, If not, just upgrade as much as possible. The trick is that the more obsolete something is, the less it costs to upgrade to. Therefore in peace years, you can research all you want, Then about a year before it all kicks off, upgrade everything, your troops will all race through the upgrades much quicker. Forts are wonderful for defence, but take to long to build, so don't bother with them.
3. Organisation
Once war begins, stop building anything, it is now too late. Maximise your reinforcements, and keep an eye on how much is needed. You will be constantly balancing repairs and upgrades. Make sure to use and remove the prioritise button too, those troops will get resources first if there isn't enough to go around. Make sure you are not wasting resources on navy or airforce upgrades too. Those will be treated equally to land troops unless you specify otherwise (simple tip, disband both! all ships in the game are useless compared to submarines and as a small nation an airforce is too expensive anyway). Make sure you appoint the best ministers, those that add to organisation or regain are what you need, if there is one for +20% regain, you are already at a massive advantage.
4. Battle tactics (general)
ok, so you are not trying to actually defeat the enemy, or advance at all, just grind him down for months or years until he cannot resist. You need to spread your troops evenly, and have them divided into smallish groups. So, if you have 12 troops per square, having two groups of six, or four groups of four is much better than a single 12. Now, attack with half the troops, letting the others recover. Then when the first lot are exhausted, swap to the second group.
5. Battle tactics (specific)
These tactics are for when your opponent is bigger and stronger than you, so just attacking him, even if you are maximising repairs, will probably not be enough. You need to have local superiority. So, lets say you have a line of defence in squares 1, 2, 3. And against them is the enemy in squares A, B, C.. The perfect situation is where you can get your troops in 1&2 to attack A. Then 1&2&3 to attack B, then 2&3 to attack C. This way you outnumber the opponent. The exact layout of the map means it's hard to get this perfect, but it is the way you need to be, making sure you can always double up, abandon any area that is isolated. Once you defeat the opponent, immediately stop attacking. You do not actually want to advance. The area being attacked will suffer damage to infrastructure, which will reduce supply and repair rates. Your area will be unaffected. Also, leaving the area empty will make sure the computer will immediately try to replace the defeated army by shuffling his resources around, so when he does this, attack again. As the computer armies arrive one by one, you should easily be able to swat them away. This will cause havoc with his plans, as he will now be constantly moving troops around, instead of attacking you. Make sure to not attack forts. Troops in forts not only get a boost to defence, but fight to the bitter end and don't retreat. If the enemy does break through, then there is no need to alter your tactics. As you are always doubled up, you should be able to attack him on either side to defeat him, then quickly return your defeated troops to plug the gap.
6. Aircraft.
in the HOI base game, ground attack aircraft were massively overpowered, and troops in movement were ridiculously vulnerable to air attack. Furthermore, such aircraft were incredibly cheap to build. Therefore it was possible to win any war (and I mean defeat Germany as Poland in 1939) by using these tactics, and annihilating the attacking (and then retreating) armies from the air. This is not the case in Darkest Hour (the overpowered unit here is heavy submarines), and so while this tactic still works it doesn't break the game. In this case, only after about 1943 are ground attack aircraft good enough to make a difference, and they aren't cheap either. However, if you do have the resources to make and use them (and repair them, and build interceptors to defend them) then they can help a lot. Select to attack a single province and say ground support, this reduces organisation, then when the army is moving, switch to ground attack.

I have used these tactics to win many times as Nationalist China, Also win as Abyssinia in 1933 AND 1936, plus defeat the USSR as British Raj (using the Korean Peninsula)