Do I need to follow the grappling rule "making a grapple check instead of an Attack roll: a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use)" when grappling an ally who is doing death saving throws? As he is an ally and wants to be grappled, he wouldn't try to escape from my grappling. Also, he is kind of dead so he shouldn't be able, should he?
There is no need for a contest.
Grappling is a contest (PHB p. 195):
Battle often involves pitting your prowess against that of your foe. Such a challenge is represented by a contest. This section includes the most common contests that require an action in combat: grappling and shoving a creature.
But contests only matter when there is opposition, according to the rules for contests (PHB p. 174):
Sometimes one character's or monster's efforts are directly opposed to another's. This can occur when both of them are trying to do the same thing and only one can succeed.... This situation also applies when one of them is trying to prevent the other one from accomplishing a goal.... In situations like these, the outcome is determined by a special form of ability check, called a contest.
The contest rules for grappling are only provided to help resolve a conflict between two opposed parties. There's no opposition in the case you present, so there's no reason to use those rules. In fact, there's no real reason the grappled condition should be necessary.
You can drag a body weighing a number of pounds up to your Strength score times 30 without any question (PHB p. 176). Your speed would be reduced to 5 feet if the person is heavier than 15 times your Strength score, which makes this mostly viable if you are strong and they are light, but even with such a penalty most characters would still be able to drag the fallen ally 10 feet per combat round by dashing, which can make the difference in some scenarios.
Alternatively, the DM might just ask you to make a Strength check of some sort to see if you can drag your unconscious ally without penalties.
So, just try to drag the body by following any instructions the DM gives you, because grappling doesn't fit the situation.
Up to the DM
There really isn't a RAW answer to this and it is very similar to a ruling Jeremy Crawford made regarding grappling and the stunned condition. While Stunned and Unconscious conditions have different mechanics, they are similar in that both creatures are Incapacitated.
The rules leave the answer to this question up to the DM. In most cases, I'd have a stunned creature fail to resist a grapple.
Approaches to Schroedinger's Body
The biggest issue here is that interacting with Creatures is different than interacting with Objects. In order to pick up an object, you only need to look at maximum carry weights. In order to pick up a creature, you generally utilize the Grapple rules.
An unconscious body is a bit of a Schroedinger's Cat. It is basically a lump of flesh, but it's still technically a creature. A DM could either rule for a Contested Check (standard grapple), uncontested (standard object), or the DM could set a DC for being able to do this.
At my table, we've ruled that picking up an conscious/unconscious ally is just a DC check(with some variable on DC based on the difficulty of the particular situation) and it's worked for us.
Crawford discusses on TwitchTv
Jeremy Crawford discusses this further on TwitchTV (at about 25:30 in the stream). He suggests using the Grapple rules, but it's unclear as to whether or not he means the half move part of them or the full contest.
When a characters are reduced to 0 HP, they fall unconscious. The unconcious condition states:
An unconscious creature is incapacitated, can’t move or speak, and is unaware of its surroundings...The creature automatically fails Strength and Dexterity Saving Throws.
Incapacitated Condition states:
An incapacitated creature can’t take actions or reactions.
Although strength (and Dexterity) checks do not cost an action or reaction, logic dictates that you have to be able to move your body. Thus he would not be able to contest the grapple. You can grab him and drag him as you wish.