Expression of a gene in different groups I would like to check the expression of a gene in different groups like Disease vs Normal samples. I want to make a plot out of that to check whether it is significant or not.

From this paper lncRNA I see that Figure 1C they used RPKM value. But, I'm using read counts data. My question is can I use log cpm to make a plot like that or should I need to use only RPKM?
You should never use RPKM. It’s simply obsolete in the age of paired-end sequencing, and has been replaced by FPKM (which is, strictly speaking, a synonym).

The linked blog post explains more generally the problems that measures such as FPKM and CPM suffer from. A more robust measure is the TPM (transcripts per million), which scales CPM by the lengths of the transcripts.

Lastly, none of these measures perform very well for comparing gene expression across replicates. Since you have read count data, you should run DESeq2 or edgeR to perform differential expression analysis across all genes, and go from there.

As the comment noted, the paper you cite does not represent scientific best practice. In fact, a competent reviewer would have rejected this analysis.