You have the tools or usable equivalents, and you should have the knowhow, but the risk of failure is high, and the result will not be a true Space Marine. In plot terms, this means that if the GM says yes, you have an almost Space Marine on your hands, as it wouldn't be particularly interesting if it failed.
What you have
A lot of the answers claim that you don't have access to the various organs that make a Space Marine. They are mistaken. All the Space Marine implanted organs have one source, the gene-seed. You can't just shove the seed into the guy, that would certainly alter him, he'd get stronger, tougher, faster, but he wouldn't even be close to a Space Marine.
What you do have is cloning vats. Space Marine organs are all cultivated from the gene-seed, that's all you need to get started, the forces of the Lost and the Damned have been making Traitor Marines with less for ten thousand years, compared to what they're doing, you've got it easy. Your ship almost certainly has the vats you need on board, and if not, you can buy them with no questions asked.
What you don't have
You have all the organs thanks to the seed, but you're missing some of the inorganic implants. The Black Carapace is a big one, however, it isn't necessary. You can do without it, try taking the one from the dead Marine you have on board, or build an equivalent yourself. I'd recommend the former or the latter, there's no need to increase the risk of failure just for the Carapace. Without it, the only effect will be that unlike a true Marine, yours will not avoid the penalties of wearing Power Armour, he'll act the same as any other power armour user.
The other inorganic ones are either not necessary, or not unique to Marines, meaning you can get a hold of or make them yourself.
First of all, there is a major issue you need to address. Your volunteer. Once your volunteer hits a certain age (I personally go with 25 for several reasons) they will no longer be a viable candidate. You can try it anyway, there are ways to make it work, such a rejuvinens or less expensive hormonal treatments, but that's adding more risk of failure to an already very risky procedure. Your ideal candidate will be around 9 years old, though as old as the early 20's should work.
The sex of the volunteer should be male, again, you can try and change this, but the risk of failure would approach 100%. That said, if this is happening, the rule of plot dictates that it will succeed, so go nuts.
Once all the organs are cultivated, you can go about the implantation process. There are better sources to find out the order and function of each organ, so I won't go into details. The full process should take years, but there are cases of far shorter augmentations, so much like most of the above, you can treat that as a guideline not a rule. A lot of that time is physical conditioning and brainwashing, both of which can be skipped.
So many things can go wrong with this that I wouldn't even attempt it unless the GM is on board and rule of cooling it, or if you have a brilliant chirurgeon aboard. To do this with rolls rather than GM fiat, you need somebody who basically can't fail medicae rolls. Totally possible to get a medic that good as a player character, but good luck finding an NPC like that if you don't already have one.
Even if it's successful, there will be problems. As I said above, your Marine won't be able to wear Power Armour as if it were a second skin, like he would if he had the Black Carapace. On top of that, the defects of the gene-seed will have an effect. You'll need to do a full autopsy on the dead Marine to check whether their chapter has any variations on the augmentation procedure. Some organs may be left out, or augmented with cybernetics, and you can't f#$k that up.
Any defects the chapter has will be present in your Marine. Almost no Space Marines have functioning remembrancers, and most don't have working Betcher's Gland's either. Any mental defects from the chapter will cause issue, the effects could be more pronounced or less pronounced due to the lack of extended brainwashing.
All but the most slapstick of complications will only be apparent after the surgery. The volunteer may die, horribly. You may be able to try again if it fails, but that's pretty sketchy. This procedure is practically mundane compared to trying it again if it fails.
Despite what it might seem, this isn't actually all that unusual. All new Traitor Marines are created in far less ideal circumstances, and provided you have a good chirurgeon, you quite literally have everything on hand that the actual Marines have, save for the Black Carapace. The odds of failure are so high due to the odds of failure for the legitimate procedure being so high, and in 40k, rule of cool means that those odds are basically irrelevant. Corruption for everybody involved is likely, but if that happens, the odds of success go up dramatically as well.
The risk of discovery by the imperium is far higher than the risk of the procedure completely failing. So I say go for it.