In modern phylogenetics there are, broadly, two approaches; morphology and genetics. Morphology involves measuring and comparing the shapes and sizes of different body parts, and genetics involves testing the genetic material (DNA) and comparing it to the genetic material of other animals. There are arguments in support of, and against, both sides. For example, just because two animals share a similar shape and size doesn't mean they're related, but the chances of two unrelated species having very similar shapes and sizes is pretty small. The same goes for genetic material; even if we tested an entire genome, we have no way of knowing how many of the resulting similarities or differences are indicative of a relationship, or just a coincidence, but coincidences would not form the majority of the results.
We can also use archaeological methods to contextualize our findings; for example, we know that modern humans were not the first hominid to use tools; in fact, we probably inherited the technology from another species. We can analyze whether a body was capable of standing upright, running, using tools, or looking at the relative size of its brain, to judge its relationship with modern humans.
The phylogenetic tree for hominids is constantly changing as we discover new material and specimens; as we discover this material, it is important that we review previously discovered material as well, to look for potential connections that may have been missed as a result of not being aware that a connection was possible. For example, if humans and Neanderthals share DNA, then it is possible that other hominids were cross-bred as well.
This question refers to phylogenetic processes. There are many ways we delineate species relation to one another. We can use genetic sampling, by taking the DNA from multiple organisms we can account for the DNA relation and determine which species are more closely related to one another.
Within the hominid group we can take traits from each species line and use those for identification purposes. For example, Neanderthals were distinguished by their large cranial region and larger body proportion to Homo sapians.
By using both the phenotypicand genetic features we know about an organism; we can begin to narrow down the choices of relation within a single group, such as hominoid.