How is Pseudomonas aeruginosa diagnosed?
Well, the answer is actually a Wikipedia search away. But from the 7th edition of the book Microbiología Médica and Todar's Online book of Bacteriology:


Nutriotional needs. Simple. Part of why Pseudomonas spp. grows everywhere is that it can metabolize a wide array of carbon and nitrogen-containing compounds. Blood agar and McConkey are enough for its growth.
Colonies. They generally show a mucoid appearence and are flat.
Respiration. Practically an obligate aerobe. Usually oxidizes glucose, but it is capable of anaerobic respiration as well, using nitrates or arginine as electron donors. The cytochrome oxidase assay is positive, unlike Enterobacteriaceae and Stenotrophomonas.
Pigmentation. P. aeruginosa synthesizes pyoverdin (fluorescent and green) and pyocyanin (blue and water-soluble). Pyocyanin is produced when the presence of iron in the media is low, however, and not all strains synthesize it.
Temperature. Its optimal temperature is 37ºC, but its survival range is quite wide, from as low as 4ºC to as high as 42ºC.
Antibiotic resistance. And to many antibiotics. The presence of beta-lactamases makes P. aeruginosa withstand penicillin, carbapenems and cephalosporins.
Odor. Cultures emit a pleasant aroma that most associate with grapes or soap.


Wikipedia does show a nice battery of tests that identify P. aeruginosa, but if I were you I would compare them to other sources.


Positive: oxidase, citrate, motility, gelatin hydrolysis, acid production (from glucose and mannitol), nitrate reduction, lipase, pigmentation (green or blue), catalase and beta-hemolysis.
Negative: Gram staining, indole production, methyl red staining, Voges-Proskaeur (?), hydrogen sulfide production, urea hydrolysis, phenylalanine deaminase, lysine decarboxylase, acid production (from lactose, maltose and sucrose) and DNAse.