Welcome to the world of lateral thinking in 4th ed. It's pretty essential for you to discuss this with your DM.
As someone who prefers to go through a wall even if there's a door available, I empathise with your plight. You'll be happy to know that after you pass mid-heroic, there are plenty of ways of getting from point a to point c without going through point B. By the time you hit epic, Primal Grove makes moving armies around remarkably cheap and easy.
This also tends to ruin 4e games. (As I have learned)
The glory of 4th edition is the set piece battle. Fighting and exploring the narrative of combat in 4e is really quite fun. Amplifying that narrative with your character's particular style is, arguably, absolutely essential to making that fun for you.
In order to do this, you must negotiate with the DM. If she knows what to expect, she can design in "breakable walls" into the scenario. (Otherwise, expect most walls to be 5 foot thick unobtanium that has 0 resale value.) If you use wall-breaking as a chance to give you a tactical advantage while getting to the set-piece battle, fantastic; if you use it as a way of avoiding the battle, you may want to play a diferent game.
It is trivial for the prepared DM to provide opportunities to shine simply by using the "breaking wall" rules. (note that they are not designed to scale, so to continue this theme in paragon you'll want to negotiate appropriate DCs with your DM.
Unfortunately, the unprepared 4e DM is likely to say no, or invent an unexpected 10d6 of gnolls on the other side of the wall you just broke. The narrative logic of 4th ed provides for islands of huge mechanical detail separated by seas of essentially pure narrative controlled entirely by the DM.
With that said, here are some options for your barbarian
Stone shape is a level 8 exploration ritual. If you take vistani heritage and vistani pathfinding, you can cast it once per day for free, even though you aren't a ritual caster. This segues neatly into Passwall at 12. For your purposes, it's fairly important to refluff all of this as a barbarian hitting the wall, but it represents ever larger "walls" you can break through.
In mid to late heroic, you can simply break through thin walls. A 6" wooden wall is a DC 25. A masonry wall (1 foot) is 35. And there are plenty of utility powers that boost those checks:
- like the Vampire's Strength of Blood.
- Crucial is the utility power Kord's Force which substitutes an athletics check for a strength check. These, of course, synergize well with Kord's Mighty Strength, which provides bonuses on just these sorts of activites.