The only trick is a backslash avoiding single quotes and surrounding everything else with single quotes:
alias p4client 'echo `p4 info | perl -ne '\''s/Client name: (.*)$/print $1/e'\''`'
Although perl can be simplified:
alias p4client 'echo `p4 info | perl -ne '\''print /Client name: (.*)/'\''`'
`` , ; ( perl , p4 ):
alias p4client 'p4 info | perl -ne '\''print /Client name: (.*)/s'\'
-, :
alias p4client "p4 info | perl -ne 'print /Client name: (.*)/s'"
, bash, , = p4client.