You can split the data by group to produce the boxplots. There are several ways to do this, I usually use split() as I've shown here, but other ways are also mentioned in the answers there.

To add the asterisks for p-values I suspect there might be a geom for this in ggplot2, but I haven't done it yet, so either someone else will chip in or I will try to look it up later. Ggplot2 is the library used to produce your example plot. There is a very good introduction on youtube.

You can use dput() on the data object and then paste the output into the post. That saves you from using pastebin.

I have manahe to do some progress, but the result is strange.....I cant tell where I'm wrong in general?

data <- read_delim("data.csv", delim = ";", 
                   escape_double = FALSE, trim_ws = TRUE)


myplot <- ggplot(data, aes(x = group, y = log2FC, fill=group)) + geom_boxplot() + ggtitle("Expr.") + theme(text=element_text(size = 16))

cmpr <- list(c("P","hCG+7"), c("P","hCG+9"))
myplot + stat_compare_means(comparisons = cmpr, tip.length=0.01,
                            label = "p.signif", 
                            symnum.args = list(cutpoints = c(0, 0.0001, 0.001, 0.01, 0.05, 1), 
                                               symbols = c("****", "***", "**", "*", "ns")))

I got this:

And warning:

Warning messages:
1: In wilcox.test.default(c(1L, 1L, 1L, 1L, 1L), c(15L, 14L, 10L, 11L,  :
  cannot compute exact p-value with ties
2: Computation failed in `stat_signif()`:
non-numeric argument to binary operator 
>