Try this:
x <- data.frame(v1 = c(2,1,1,2), v2 = c(1,1,3,2))
# The order function returns the index (address) of the desired order
# of the examined object rows
orderlist<- order(x$v2, x$v1)
# So to get the position of each row in the index, you can do a grep
x$rank<-sapply(1:nrow(x), function(x) grep(paste0("^",x,"$"), orderlist ) )
x
# For a little bit more general case
# With one tie
x <- data.frame(v1 = c(2,1,1,2,2), v2 = c(1,1,3,2,2))
x$rankv2<-rank(x$v2)
x$rankv1<-rank(x$v1)
orderlist<- order(x$rankv2, x$rankv1)
orderlist
#This rank would not be appropriate
x$rank<-sapply(1:nrow(x), function(x) grep(paste0("^",x,"$"), orderlist ) )
#there are ties
grep(T,duplicated(x$rankv2,x$rankv1) )
# Example for only one tie
makeTieRank<-mean(x[which(x[,"rankv2"] %in% x[grep(T,duplicated(x$rankv2,x$rankv1) ),][,c("rankv2")] &
x[,"rankv1"] %in% x[grep(T,duplicated(x$rankv2,x$rankv1) ),][,c("rankv1")]),]$rank)
x[which(x[,"rankv2"] %in% x[grep(T,duplicated(x$rankv2,x$rankv1) ),][,c("rankv2")] &
x[,"rankv1"] %in% x[grep(T,duplicated(x$rankv2,x$rankv1) ),][,c("rankv1")]),]$rank<-makeTieRank
x