So there are a number of individual issues that have to be addressed, each one with its own set of plausible solutions.
1. Long-Term Goals
There is the question of ruling forever, but will this king want to? When you are basically ageless, the perspective of time most likely changes. So in say 300 or 400 years, will the king still want to rule? Can the feudal monarchies in your world survive that long and will he want to be a part of the political revolution that follows or at point will retiring filthy rich to an island be the goal of the day? That or sneaking out to avoid the revolution
It may not be part of the question directly, but the length of time he plans to rule combined with his personality will influence what kind of plans and preparation he will be willing to undertake to maintain his rule.
Is taking a few decades off and letting another house rule in order to travel to pursue knowledge part of his plans or will he want to be on that throne for as long as he is able to be?
2. Laws against Vampirism
In the low-fantasy setting I'm working on, the king is a vampire in a world in which vampirism is punished by death.
Who has made this law? The global religious order, the king (or a previous one), or a coalition of monarchs?
Depending on the monarchy style, then the king will have the power to arbitrarily change the laws to suit his needs. Of course there are the lower nobles that administrate his lands and might object to such a radical change of laws through revolution, or accusations of vampirism because of such a sudden and drastic change in laws. So slowly changing the laws are still the safer order of the day regardless of his effective power. Should the king's laws have to go through a parliment that has actual power, then it is trickier, but again the slower approach has a higher chance of success.
The end goal would be legalization of vampirism outright, but even arranging a change of laws so that vampirism is not outright illegal while a vampire provably killing somebody is punishable by death is a lot of breathing room because at that point, the law can't kill your for existing. At least not directly.
3. Blood Supply
Vampires have a strong desire to drink human blood, and once they start, they usually cannot stop without killing their victim. It's not a biological necessity, but they suffer ever-increasing withdrawal symptoms if they cannot drink enough blood often enough.
The key word here is "usually" -- it implies that with enough self-control, a vampire can end up not killing their victims. If the king understands his condition enough to know how often the desire needs to be fed, and how much blood can appease it, then he could theoretically keep it fed to a point where a few most trusted servants are fed upon nonlethally in their sleep. However, that would lead to people questioning things which could be a Bad Thing if he is caught or a retainer objects.
Another possiblity not previously mentioned is that the king moonlights as an executioner or acts as one under special circumstances. Death by exsangunation for feeding purposes followed by a swift beheading to hide the evidence deals with both condemned criminals as well as the king's hunger. The beheading also conveniently prevents the now dead body from rising and if chopped in the corrct place can hide the fact that there are/were fang marks on the body.
Naturally, the king's executions are done in private. The official reason is to keep certain criminals from gaining fame and popularity at the headsman's block as a martyr. An alternative reason is that the condemned do not deserve a public death. The real reason is of course to hide the blood feeding before the actual execution.
An interesting thought might be that the king is the last point of appeal with a lethal twist: The loser of the appeal has a date with the king's fangs in private for wasting his time. This will only be for people making the appeals -- it would have no bearing on items brought to the monarch directly.
As stated, a more modern king will have access to a blood bank for a rare (and heriditary) medical condition. While it not fresh from the vein, it is still human blood, and so long as he is not demanding the best of the best, there should be no problem using the most common blood or something equally plausible.
4. Avoiding Suspicion
[P]eople are going to find it odd after a few decades when they notice their king doesn't age. The commoners are easy to fool, they never meet the king anyways. But the highest magistrates are another story.
This is the larger problem and has been covered better by other people so I won't really touch this concern too deeply.
One thing I will observe is with enough time, your king will more likely want to be known as a sagacious ruler as opposed to a battlefield conquerer. He will have had the potentially extra decades to accumulate knowledge and wisdom. This is not to say that he will shy from war when needed, just that it will not be his primary method of consolidating power. Also, a war might even be a good time to plot faking his own death and being his own successor.
Makeup and hair dye will be the kings best friend in appearing older than he is. That and acting skills. As there are no special benefits to his immorality aside from the agelessness, then that should be sufficient.
If he has perfect recovery, as in he is immune (or highly resistant) to infections and other non-violent means of sickness or death, then extoling that as a divine blessing or gift could be the way to go.