First, fanaticize your population.
First, denounce your enemy by creating an us-vs-then mentality. I wrote a lot of possible topics for this in this answer which I could all repeat here. tl;dr version: Convince your population that the population of the other country are unholy heretics, have no morals, are culturally unrefined and their leader is an idiot.
How could you do that?
In medieval times, there were no newspapers. The people had three information sources about what happened in the world:
- Public announcements by government officials during market.
- What they got told during mass at the church
Having the kings lackeys go around taverns and spread rumors might not be very efficient and scalable, but it is very easy for your king to maintain plausible deniability. After all, even he can not control what his lackeys blabber about in a drunken stupor.
Paying some market criers to announce their agenda should be under the direct control of your king, but it is a very obvious method of influencing the public opinion. When the king wants to avoid suspicion that his goal is to start a war, their messages must be subtle.
The church might or might not be able to be persuaded to help - it depends. In medieval times, the relationship between nobility and clergy was complex. It was often quite unclear which had authority over the other in which matters. Sometimes the church cooperated with the plans of the government, and sometimes they didn't. When dealing with Catholic clergymen, your king has to remember that they are not completely autonomous but have to answer to the pope (or currently popular antipope). But when your king can persuade the clergy to publicly denounce the population of the other country as heretics, it would be an immensely powerful tool to manipulate the masses. To avoid suspicion, the king should make it look as if this was the clergies idea, not his.
Then, make them afraid
When the population is sufficiently convinced that they are the good guys and the other country is the evilness which embodies everything they don't embody, make your population believe that the other country is a direct threat to them. Use the tools listed above to tell them about actions the other county has already taken against their well-being (framing them is also a good excuse for any problems you created yourself through bad rulership). Also drop hints that they might be planning a large-scale invasion.
Finally, start your crusade
Now that your population believes that the evil will destroy them soon, most of them will support you when you speak of taking military actions. Volunteers will line up at your garrisons. With the current climate, you can easily denounce any pacifists - among the peasants and also among your council - as collaborators and traitors and have them executed. This will make people think twice about opposing you.
But my king is too lazy and stupid for all of this
The secret of efficient leadership is delegation. No king can manage a whole kingdom all by themselves. When your king's father taught him anything at all, it should be that. When he hasn't got the time and talent to execute this plan or even come up with it, he might have a clever and diligent subordinate to do it for him.
Take King Henry VIII, for example. Without people like Thomas Wolsey to get stuff done for him, he would have just been a footnote in history. While Henry VIII spend most of the first decade of his regency fooling around, Wolsey acted in his name and achieved more for him than most actual kings achieved in their whole lifetime.
A proper motivation for that subordinate could be a promise to become the lord of the newly conquered lands. This should get the loyal subordinate a council position which looks like a win-win situation for the king (a sufficiently clever subordinate will likely do a 180° turn after actually getting the position, but your king might be too naive to realize that).