I used to believe the Dwarves were among the worst kingdoms, if not the worst, so I tried them. My opinion changed immediately!
1) Gold - so much gold! And good food production, too!
I suggest starting a pair of 600 standing orders - one for gold and the other for food - at the capital as soon as possible, even on turn one. These orders pay off for the DW quicker than most kingdoms (raising gold x 3 costs 6750 gold; the DW gets +3300 gold production from a single order). When you are comfortable, pair that with a Dwarven raise defense standing order, and you will quickly have a solid capital. Be sure to upgrade that town to a fortress, of course. If you start on turn one, then by turn five, your capital will produce an extra 16500 gold (for the cost of 33750 gold).
A word about ESO - the DW is one of the kingdoms where having the ESO requirement of a PC with 35k production or defense is fairly trivial.
Don't neglect setting up your gold mine. I didn't set mine up until turn five or six and was kicking myself. It may not seem like much at first glance, but an extra 20k gold per turn, unaffected by seasons, for the rest of the game is an excellent kingdom perk.
Depending on which region you began in, you should be looking at around 140k gold and 80k food production on turn five (less if in winter, of course). A lot of kingdoms would like to be in that spot on turn five.
2) Attack a few villages
You are correct; the requirement for elite troops bottlenecks the DW, so you must work toward your first elite brigade ASAP. The first step is getting veteran troops. Some can be had by regional selection - Zanthia, Mythgar, and others provide veteran troops. Also, I suggest considering using the customization order to upgrade a kingdom brigade. Beyond that, you will need to attack stuff!
Early in the game, it is typical to discover human/neutral villages in your area, which people typically take with emissaries. Use your military instead to promote leaders and possibly troops. You may want to do a pop search (order 725) with the 1DW in hopes of finding a village right away. About a 3:1 to 3.5:1 advantage seems to work for me, where the troops can take the village, get a little bloody, and also get a promotion. Depending on the setup, you should be able to find one or two villages to help advance your military. Don't worry about losses. If you stick to villages and have triple the defense value, you are very unlikely to lose a brigade. Attrition can be addressed later post Wildling recruitment.
3) Find a few encounters
If possible, use encounters to advance your leaders. The Dwarves have both the Adventurer and Heroic traits, making them adept at investigating encounters and getting promotions. Using the customization order to locate the fine encounter in your home region should be a consideration.
4) Plan for turn 10
Turn five, you will have a region and lots of production. Decide what you want your turn 10 to look like. Do you want several p-3 wizards? It's always good to have. Do you want a strong group of Dwarves, Wildlings, and War Machines? Do you want to recruit troops from pop centers and train them to get their bonuses (thinking Zamorans and Westmen here)? Do you want a collection of fortresses around your towns? Some mix of all of the above? Whatever it is, plan for it. Few people want to attack the DW early, so use that time to plan what you want your kingdom to look like when coming out of the early game and into the mid-game. You will get turn ten reinforcements plus, ideally, an ESO reward; plan for those resources. I'm sure you have at least one neighbor you will want to attack
5) Combat
Some things to keep in mind when it comes to combat. Versus PCs the Dwarves are powerful, especially with veteran siege engines. Once the DW gets going he can smash PCs easily. Versus other groups, the DW has several advantages - immune to fear and chaos, immune to death magic and takes less damage from magic overall. If you have a warlord (and you should by the early teens), you are also immune to flanking attacks. Those are a lot of advantages. Keep those in mind when facing enemy groups.
6) Mithril
Don't neglect mithril. Mithril upgrades turn DW troops into monsters.
Hope this helps!
Buddy/Ignore List
