Many people think that Tibet is not accessible from Nepal. The main reason may be the way Tibet is portrayed in the movies and even in news.
It is also not true that the northern border of Nepal, between Nepal and Tibet (China) there are only big Himalayas.
There are enough passes to move from Nepal to Tibet. The best one for trekking to Tibet from Nepal is through Rasuwa.
Rasuwa district lies around 130 kilometers on the north of Kathmandu. There are daily buses that go from Kathmandu to Rasuwa, if you wish you can take a private vehicle as well.
From Rasuwa, you can just walk a few hundred meters and enter Tibet. However, you need to have an organized tour to travel to Tibet
The tour can be organized by travel agencies in Tibet. If you call them a few weeks before you actually want to get to Tibet, they will arrange for your passes plus the itinerary.