The easiest way to gain advantage on a death saving throw is to use inspiration.

If you have inspiration, you can expend it when you make an attack roll, saving throw, or ability check. Spending your inspiration gives you advantage on that roll.

Alternatively, you could be a Wild Magic Sorcerer, and use Tides of Chaos. Well, probably.

Starting at 1st level, you can manipulate the forces of chance and chaos to gain advantage on one attack roll, ability check, or saving throw.

The 3rd- level spell Beacon of Hope explicitly grants advantage on death saving throws for its duration.

Fighters can use Indomitable to reroll a failed saving throw - not precisely the same as advantage, but pretty close.

Beginning at 9th level, you can reroll a saving throw that you fail.

There are almost certainly more options that I've missed - getting advantage on things is a pretty fundamental part of 5e.

Dwarves do not have advantage against the Constitution saving throw induced by Contagion.

This is the relevant text in one place to make it easier to answer your question. The description of the Contagion spell (PHB p. 227, per PHB Errata p. 3) says:

Your touch inflicts disease. Make a melee spell attack against a creature within your reach. On a hit, the target is poisoned.

At the end of each of the poisoned target’s turns, the target must make a Constitution saving throw. If the target succeeds on three of these saves, it is no longer poisoned, and the spell ends. If the target fails three of these saves, the target is no longer poisoned, but choose one of the diseases below. The target is subjected to the chosen disease for the spell’s duration.

Since this spell induces a natural disease in its target, any effect that removes a disease or otherwise ameliorates a disease's effects apply to it.

1. Your touch inflicts disease.

2. On a hit, the target is poisoned.

3. At the end of each turn you make a Constitution saving throw. The Constitution saving throw does not specify the poison as the origin.

4. 1 If you succeed three Constitution saving throws, then you are no longer poisoned and, the spell ends. The target is subjected to the chosen disease for the spell’s duration, so the disease ends.

5. 2 If you fail three Constitution saving throws, then the poison condition ends, and you are subject to the chosen disease.

To be under the effect of a disease, the caster has to choose one of the diseases.

Dwarven Resilience, PHB p. 20:

You have advantage on saving throws against poison

So as a Dwarf you do not have advantage against the Constitution saving throw, as it is not a saving throw against poison or taking poison damage, and you are normally affected by the disease, until the end of step 4.1 or until the disease conditions of step 4.2 resolve.

In similar situations when considering poison damage or poison as a condition, the DC against that poison is most explicitly worded as save against poison, or as save or take poison damage.

Poison, Basic, PHB 153

A creature hit by the Poisoned weapon or Ammunition must make a DC 10 Constitution saving throw or take 1d4 poison damage.

Poison Spray, PHB 266.

The creature must succeed on a Constitution saving throw or take 1d12 poison damage.

Ray of Sickness, PHB 271:

On a hit, the target takes 2d8 poison damage and must make a Constitution saving throw. On a failed save, it is also poisoned until the end of your next turn.

Stinking Cloud, PHB 278

Each creature that is completely within the cloud at the start of its turn must make a Constitution saving throw against poison.

Giant Spider, PHB 306:

the target must make a DC 11 Constitution saving throw, taking 9 (2d8) poison damage on a failed save, or half as much damage on a successful one. If the poison damage reduces the target to 0 hit points, the target is stable but poisoned for 1 hour

Carrion Crawler, MM 37:

and the target must succeed on a DC 13 Constitution saving throw or be poisoned for 1 minute. Until this poison ends, the target is paralyzed. The target can repeat the saving throw at the end of each of its turns, ending the poison on itself on a success.

This lack of wording that directly matches the poison against the DC implies that the Constitution saving throw induced by Contagion is not a saving throw against the poison, but the effect of the poison is closely tied to the condition of the disease settling much like the side effect of a fever that protects you against a disease.

This line of thought is further supported in that you make saving throws until you succumb to the longer effects of the disease or until you shake it off, including the side effect of the poison condition. The secondary effect that is the poison condition ends when you manage to save against the disease (succeed on your three Constitution saving throws) or succumb to the disease (fail on your three Constitution saving throws).

It might seem obvious that Dwarves would have advantage, especially since creatures immune to poison are immune to any effects of Contagion according to this thread based on this reply by Jeremy Crawford to a tweet. But this is a misleading assumption because poison immunity saves through the condition not being applied as it becomes an invalid target, not because you make any saving throws against poison.

Thank you to Gandalfmeansme for catching an inconsistent wording.