I'm now level 19 and invested most perks in stealth and one-handed-combat. The problem is that as
an assassin (backstabbing), I'm not gaining much experience in 1h. But I need those levels, in order
to unlock perks. My strategy so far is buying 1h combat levels with a trainer. Unfortunately, It becomes
rather expensive. Any tips of experienced assassins would be welcome

Hit stuff with a 1 handed weapon or pay a trainer.

If you want.. get the spell that lets you heal a target. Find a weakling with good HP and beat/heal rinse and repeat. I dual wield 1 handed weapons and my skill is just fine. Not sure what else to say.

What I do on my assassin like character is backstab (or sneak attack with an arrow) when I find it a viable option instead of just hacking the enemy to pieces. I have a level 22 guy and his sneak is just below 60, archery is mid 50s and one handed is like 73. I play on expert and I find its a very solid build. Just an idea of something you could try.

Go to the guardian stone and change your power to the Warrior Sign which speeds leveling of combat skills. I'm having the same problem and I'm around the same level as you, but that has helped some. Other than that and using trainers I have no idea. At this point, if I don't get a backstab, I get killed easily. My sneak is 62 and 1 hand is 47. All other skills are lower.

Also playing the sneaky sneaky Argonian.
In the one handed tree I only really focused on Dual Weild on the far right. Otherwise the perk in stealth tree for daggers and backstabbing really make the spec what it is. Those instant kills on humanoids never..EVER get old for me.
HOWEVER taking the perk for bonus damage and faster attack for dualweilds really come in handy with the dragons. Remember though; stay light on the feet and learn to dance! And all those bee stings on that dragons rump will bring it down!

First I'd find the stone that increases combat skill learning, then I'd stock up on healing potions and hit a random cave without using stealth or bow. Also, I'd use fast weapons. Save early, save often.

Thanks for the tips! ... I already considered activating the warrior guardian stone. Then just play a while as a warrior, just until I hit the minimum level I need for the required perks... Don't like this, but don't see much other options.