I want to travel to Kashmir from Europe. I know that I can easily go to Russia, but what next? What will be the route from there? If I go to Pakistani-administered Kashmir, will I be able to cross the line of control to the Indian administered part, and vice versa?
The historic principality of Jammu and Kashmir is split between the Pakistani Azad Kashmir ('free Kashmir') and the Indian state of Jammu and Kashmir. India and Pakistan have fought wars over the control of Kashmir and you can absolutely not cross the line of control which serves as the de facto border.
Not only that, but as a foreigner you are are not even allowed to enter Azad Kashmir although with a little luck you can still do so. Pakistani-controlled Kashmir also suffered from a devastating earthquake in 2005 and even if you do manage to enter, be aware that it will not have completely recovered.
Travel in Indian-controlled Kashmir is possible, if somewhat restricted due to violence between the Indian military and Kashmiris who do not wish to be part of India.
Getting to either Kashmir from Europe is fairly straightforward and there are two obvious routes, the northern and the southern. Note that the 'middle' Caucasus route is not viable due to war, closed borders and general instability in the north Caucasus.
North: if you are in Russia, then you must travel to China via Mongolia or, preferably, Kazakhstan. China's Xinjiang province borders Pakistan and the two are connected by the Karakorum highway. Parts of the Karakorum run parallel to the border of Azad Kashmir, so it is a convenient route if that is your destination. You can also follow the Karakorum to Islamabad and from there go to Lahore, which is next to the border crossing between Pakistan and India.
South: starting in Turkey, travel through Iran to Zahedan and cross the border to Quetta in Pakistan. From there you can make the long journey to Lahore (for India) or Islamabad (for Azad Kashmir).
Getting into Azad Kashmir is possible if you avoid detection by the authorities at the internal border. There are minibuses which travel from the towns Mansehra and Murree to Muzaffarabad (the capital of Azad Kashmir) and if no one sees you at the checkpoint, you are in. Even so, hotels will immediately report the arrival of a foreigner to the police, so you should make good use of your time before they come to kick you out!
Many tourists in India visit Srinagar, the summer capital of Jammu and Kashmir, so getting there is easy enough.
Your options are legion. It's more a matter of how challenging you want to make it to yourself and how much time you want to spend.
Assuming you refer to western Europe as your starting point, your easiest route would be:
Western Europe - south east Europe - Turkey - Iran - Pakistan (though southern Pakistan can be unsafe).
If you want to go through Russia:
Ukraine - Russia - Kazakhstan - China - Pakistan. Ukraine - Russia - Mongolia - China - Pakistan.
For any of the above routes, you could also detour through the Caucasus, pass through Belarus, go through central Asia and try Afghanistan.
I think that as an outsider you can cross the line of control. But I'll let someone else answer that.