Would they work like some kind of police force?

Although the Varangian Guard fulfilled a number of different functions for the Emperor, I think that it is difficult to see them as a "police force". A police force would imply that the Varangians were some kind of metropolitan guard, which is not really the role that they fulfilled. If you read the primary sources, you will not read about Varangians performing "guard duty" on street corners, but will find that Byzantine authors often talk about their raucous behavior in the city's taverns. For this, they were humorously referred to as "the Emperor's winebags".

The primary role of the Varangians, referred to by Byzantines as the "axe wielding guard", was to act as the personal retinue of the Emperor. In this capacity, they guarded the Emperor at his Imperial palace, and accompanied him as his elite bodyguard when on campaign. They were praised as his most loyal troops, but were really more loyal to the position of the Emperor than the actual person. For example, when Nikephoros II Phokas was murdered in his bedroom by John Tzimiskes, the Varangians, seeing their dead Emperor, allowed John to be proclaimed as the new Emperor instead of killing him.

Vladimir I sent 6000 to Constantinople, but would it change over time?

Certainly. Although the original contingent numbered some 6000 strong, the Byzantines eventually found other sources of "axe-wielding barbarians" in other locations, namely, Scandinavia, and Anglo-Saxon England, and recruited them into the ranks of the Guard.

If they were in the thousands, every single one couldn't protect the emperor at all time, so I wonder if they would patrol the city or crush riots (like a function in Crusader Kings 2)?

There is a lot of evidence to suggest that the Varangians were highly organized, in that they had smaller regiments and squads within the guard which would be assigned to different functions (such as: guarding the Emperor, patrolling the palace, investigating a palace plot, etc.). The leader of the Varangians in most periods was the position of Akolouthos, the "attendant", who was generally an indigenous Byzantine, but some of the Varangian runestones tell us that certain individuals were "captains of the guard", which supports the idea that there must have been a system of ranks within the guard as there was within the rest of the Imperial army.

Also, I read in an article (swe) that they wore azureblue silk clothes, is there any truth to that? In that case would it work like some kind of uniform?

The only general thing that Byzantine authors mention about the appearance of the Varangians is the fact that they wielded the famous two-handed battleaxes. When descriptions of the Guard are made, there is generally little in the way of the details of their appearance. However, the Madrid Skylitzes manuscript illustrates in several points a number of Varangian soldiers, which gives us an idea of what they may have looked like. As shown here, we see that the Varangians are depicted in their civilian attire with beards and variously colored tunics. However, if we look at this depiction we see the Varangians (top) in their full military equipment, and although most of the ink has been washed away, we can clearly see that the illustrator painted them all with blue tunics. Several other depictions/descriptions from both Muslim sources (Harun ibn Yahya), and church art (St. Theodore), also support this notion.