You can try to find someone else with a claim on the land (like a son or daughter), give them some land (either a county you've conquered once your a duke, or an extra barony), then push their claim. Alternately, you could marry a woman with a claim so your heir inherits her claim or push her claim so your heir inherits the land directly. Going further, if one of the rulers has a daughter, you could try to marry her to your heir, then murder all of her brothers so that when she inherits, her heir will be in your line of succession.
If you have Jade Dragon, the new De Jure Duchy war, Unlawful Conquest, and Force Vassalization CBs can really help speed things up.
You can also try converting to another religion. Most other religions will have a Holy War CB to conquer an entire duchy without a claim, though other Christian rulers can join to help the defender. Pagans and Muslims also have access to a County Conquest CB, where they can conquer a single county without a claim and other rulers can't join in without an alliance.
Normally, fabricating a claim will give you a claim on the county. Sometimes, it can give a claim on the duchy instead. With the ambition, it can also rarely give a claim on the entire kingdom. However, I think both of these require the title to already exist. It would be kind of pointless to fabricate a claim on the Kingdom of Ireland if there isn't a Kingdom of Ireland yet.
2. For marriage, to get a claim, just look for a daughter that has a claim on the land, cursor over it and make sure the claim can be inherited, then marry her to your heir. Now, your heir's heir will get the claim, or you can press her claim so she becomes Queen, then the shared heir will inherit the land. For expanding by marriage without war, marry someone in the line of succession, then murder everyone else ahead of them. If you click the title, you can see the current heir and the top 3 pretenders. Kill them all.
3. The two main ways of getting more levies out of your demesne are increasing your Martial and upgrading your holdings. Your martial score gives a percentage increase to all of your levies. Most of your buildings in your holdings with either offer specific troops to your levies or give a percentage increase to levy size. Also note that the army size on the character sheet includes vassal levies, retinues, hired mercenaries/holy orders, and event troops. Finally note that you can increase your levies per county by building additional holdings. This is rather expensive, and the extra holding takes an extra demesne slot or needs to be given to a vassal, but it can be effective if you don't have more buildings to build.