how can we know how closely or distantly related fossil organisms are to living organisms?

Answer is looking at the physical similarities and differences. 

Explanation: Like structural homologies, similarities between biological molecules can reflect shared evolutionary ancestry. At the most basic level, all living organisms share:

• • The same genetic material (DNA)
• • The same, or highly similar, genetic codes
• • The same basic process of gene expression (transcription and translation)

• These shared features suggest that all living things are descended from a common ancestor, and that this ancestor had DNA as its genetic material, used the genetic code, and expressed its genes by transcription and translation. Present-day organisms all share these features because they were "inherited" from the ancestor (and because any big changes in this basic machinery would have broken the basic functionality of cells).

Although they're great for establishing the common origins of life, features like having DNA or carrying out transcription and translation are not so useful for figuring out how related particular organisms are. If we want to determine which organisms in a group are most closely related, we need to use different types of molecular features, such as the nucleotide sequences of genes.