There are lots of standard methods available for determining such heavy metals. However, I would prefer the instrumental methods of Cold Vapor AAS or ICP instrumental methods for analysing those heavy metals. Since mercury is volatile, I suggest you use different instrument for analysing specific metal elements. Overall, instrumental analysis/method would fit such request.

For Cu i usually recommend Flame absorption spectroscopy, because Cu levels are usually high (in the range of mg/kg)

For Pb and Cr we use Graphite Furnace atomic absorption spectroscopy (high sensitivity and specificity) or ICP-MS

For AS, Hg, we use hydride generation with AAS

For analysis of heavy metal like Cu, Cr, Pb, As and Hg from fish tissue, You should analyze in ICP instrument for getting accurate result.

AAS and ICP-MS  methods are very appropriate to analyse heavy metals in fish tissue.

First: dissolve in  HNO3, in cool. Retain a portion for Hg (Cold vapour technique). For the other metals, heat to low volumen and use AA , , AAET or ICP..

Graphite furnace method is more versatile method for heavy metal analysis in AAS instrument except Hg . You can follow the cook book ....method is given in authentic manner, on the other way ICP - AES / OES / MS  is more useful to analyze heavy metal analysis.

after  digestion in proper solvent (generaly in HNO3), make appropriate dilution. After tis you can apply Graphite Furnace atomic absorption spectroscopy as it is sensitive and simple to use istrument, or you may also go for ICP-MS for differential analysis of the ionic form of metal too. You can also use latest generation AAS ffor it. 

In the case of Hg is very usefull the direct mesaurment, you can found as EPA method 7473. In this method don't need digestion so there is less posibility of lossing Hg