The short answer is yes, you can make your way to the higher class.
But keep in mind, aristocracy and nobility are different.
You need money to be an aristocrat. But to be a noble, you need a tittle which can't be bought, but had to be taken by force with a cassus belli or be appointed.
To become an aristocrat, you just have to bought 3 parcel of land and build a settlement. With time, it'll become an hamlet and you are officialy the mayor. Congratulation, you are an aristocrat.
Hamlet: 3 Parcels; 10 people; Well
Village: 9 Parcels; 25 people; Tavern
Town: 25 parcels; 75 people; Town Hall
City: 49 parcels; 150 people; Courthouse
Capital: 81 parcels; 250 people; Monument
To become a noble, the king can appoint you a duchy or a county, or the duke a county. But you'll need to be worthy and have relationship. Will you succeed ? Well, if they are you friend, it's easy, if not, work for it.
But an aristocrat can claim the county where he reside if enough aristocrat or noble follow him.
Check how here : https://chroniclesofelyria.com/forum/topic/6549/dj-18-kingdom-land-management
I hope it'll help you.