Cluster identification with NN
These particles build individual clusters which are already identified
  and labeled; meaning that every particle is already assigned to its
  correct cluster (this assignment is done by a density estimation but
  for my purpose not that relevant).
  the challenge is now to build a network which does this clustering
  after learning from the huge data.

Sounds pretty much like a classification problem to me. Images themselves can build clusters in their image space (e.g. a vector space of dimension width * height * RGB).

since this is not a classification problem but more a identification
  of clusters-challenge what kind of neural network should i use?

You have data of coordinates, you have labels. Start with a simple fully connected single/multi-layer-perceptron i.e. vanilla NN, with as many outputs as number of clusters and softmax-activation function.
There are tons of blogs and tutorials for Deep Learning libraries like keras out there in the internet.
I have a dataframe containing the coordinates of millions of particles which I want to use to train a Neural network. These particles build individual clusters which are already identified and labeled; meaning that every particle is already assigned to its correct cluster (this assignment is done by a density estimation but for my purpose not that relevant).
the challenge is now to build a network which does this clustering after learning from the huge data. there are also a few more features in the dataframe like clustersize, amount of particles in a cluster etc. 
since this is not a classification problem but more a identification of clusters-challenge what kind of neural network should i use? I have also problems to build this network: for example a CNN which classifies wheather there is a dog or cat in the picture, the output is obviously binary. so also the last layer just consists of two outputs which represent the probability for being 1 or 0. But how can I implement the last layer when I want to identify clusters? 
during my research I heard about self organizing maps. would these networks do the job?
thank you