As noted here, diseases in 4e seem more trivial than I think they were intended: they rarely last more than one increment!
What rules can I make/break, or what actions can I take, so that contracting a disease is actually relevant or interesting?
If you are willing to start replacing rule subsystems to improve the drama, then you could take a leaf from the FATE Fractal (see more) and model suffering from the disease with a skill challenge.
Model as Skill Challenge
Since the default rules for skill challenges aren't as good as they could be, try these alternative rules. Diseases should be a complexity 1 challenge under this system.
During play, the skills used would represent the various actions the group takes to deal with the disease (whether it has infected one party member or multiple members). This could range from hunting special herbs with Nature to using Diplomacy to persuade an innkeeper to use a room as a sickroom.
Each failure in the skill challenge would move the infected one step closer to the final state. If the skill challenge fails as a whole - the infected suffers the final state and the disease fades over time. If the skill challenge succeeds, the infected suffers the worst state they reached for a short time and the disease disappears quickly.
Each disease should be different. Creating skill challenges allows you to go into more detail about the struggle with disease and make it more interesting than a few Endurance rolls.
By no means should a GM attempt to implement many all of the following suggestions simultaneously without carefully considering the implications; some of these ideas are brutal enough all on their own.
One of the 'problems' with the disease mechanic is that "an ally can use a Heal check in place of your Endurance check to help you recover from a disease" (DMG 49). This means the entire party's defense against disease is as strong at the party's highest Heal check. The one exception, ironically, is the PC with that Heal check: he is not his own ally and must rely on Endurance to recover.
A PC gunning for max in a skill might easily have (1/2 level) + (ability modifier) + 2 (racial) + 2 (theme) + 2 (background) + 5 (training) + 2 (feat) + (enhancement bonus), and then be able to add a primary stat using an encounter power like
Difficulty Class targets have changed over the course of 4e, compensating for power inflation (backgrounds and themes were not part of the original game balance). This is the current DC table; it can also be found in the Rules Compendium on page 126. If you want to challenge skill-conscious PCs at all, go for the Hard DC to recover and the Medium DC to maintain.
If you want to be more hands-on than using a table, look at your PCs' relevant modifiers and their skill-boosting powers. Take their modifier and add 10 to make a DC they'll hit just over half the time. If the party regularly uses Aid Another, take this into account.
Ignore the Monster Attack save
If a PC is infected by a monster's attack, rather than by any other vector, she gets to make a save at the end of the encounter. On a success, the disease is sloughed off immediately. I strongly suggest you ignore this: it turns disease into an encounter-long debuff instead of a long-term challenge, which in turn encourages strangely imbalanced disease designs with weird long-term implications.
Change the track's starting position
By far the single greatest contributor to the trivializing of all disease is that most of them start at position 1 on the disease track. This, combined with the fact that position 1 effects are usually minimal, means that a single successful check will cure it before it becomes a burden. You could make position 1 effects more burdensome, but that still doesn't change the fact that combined with the points above the disease never lasts long enough to make a narrative impression.
So instead, just drop the disease's starting position to 2. This ensures it'll last long enough to be noticed. (If the disease is one of the short-duration ones with "dies" as position 3, you'll have to invent a new position 3 and shove death over to position 4, or you may wind up with an entirely different kind of disease problem --especially if you've also pumped up the DCs.)