After cleavage I know no good way to remove the reamining amino acids. The best way would be to remove them by mutagenesis in the plasmid construct but do you really need to remove them ? (do they interfere with your protein function in any way?)because if there is only 10-15 bp between his tag and your protein that means max 5 amino acids between you tag and your protein so after cleavage there should be 1 or 2 remaining, so of course it depends  what you want to do afterward with the protein but I'm not sure it is worth doing it. The other way  would be to purify without any tag...

The first question that arises in my mind is whether the vector you have chosen to clone your gene of interest in has a protease cleavage site, such as an enterokinase cleavage site, immediately following the HIS tag. For some commercial vectors, this is a common characteristic. If there is not such a cleavage site, you can always clone one in/add one to your gene of interest using PCR primers. If you put the protease cleavage site in frame with the HIS tag and your gene of interest, you can probably remove the majority of the amino acids that connect the HIS tag to your protein. I take it you may already know this stuff, but from your question, it is difficult to know where you are starting product/knowledge wise.

After cleavage I know no good way to remove the reamining amino acids. The best way would be to remove them by mutagenesis in the plasmid construct but do you really need to remove them ? (do they interfere with your protein function in any way?)because if there is only 10-15 bp between his tag and your protein that means max 5 amino acids between you tag and your protein so after cleavage there should be 1 or 2 remaining, so of course it depends  what you want to do afterward with the protein but I'm not sure it is worth doing it. The other way  would be to purify without any tag...

The vector has already a ckeavage site.In addition  I was thinking of add the His6 in my primer. Do I need to add a protease cleavage site after the tag and before my gene of interest? because in this case, bearing in mind that we need approximately 15-21bp from our gene, the primer should be around 40-50bp in length. is it possible?

in the second scenario, the vector does not contain any tag or cleavage site

gel filtration can help you.run a gel filtration after cleavage

For NO amino acids at the (cleaved) n-terminus of your protein the only real candidate that I know of is SUMO that is present with an N-His in many vectors and when cleaved with SUMO protease leaves no n-terminal residues on your protein.

For TEV cleavage you are left with an N-treminal G or S (depending on plasmid used) and with 3C/Prescission you are left with an n-terminal GP.

However if you use restriction/ligation for cloning you will almost certainly be adding extra codons in the form of the restriction sites and the only real way around this is to use In-Fusion or SLIC-based cloning vectors (e.g. pOPIN suite from OPPF-UK) OR add the whole N-term tag+spacer+cleavage site by (nested) or just spacer+cleavage site into an existing N-his vector by PCR as you have suggested.

e.g. SSGLEVLFQ!GP (spacer-3C) would be 33 bases (plus another 10 at 5' for restriction site and 20 for priming on your gene=63) could easily be made with two overlapping 45-mer forward primers.

I'd like to thank you all for your response. After all, I'm thinking to perform the cloning using the In Fusion System and a vector that I already have in tha laboratory. I believe is one of the systems with better eficasy in removing undesirable residues. However, I've never worked with overlap PCR. Does anyone has any guideline to send me? The primer I'd like to design has the 6His tag, the Cleavage Site and approximately 20-25bp from the desired gene.