Well, to begin with, five weeks is a very short time to really give the chance to the two cats to get to know each other.
Usually when you adopt a second cat, you should introduce the cats very slowly, over a period of several weeks, a little at a time.
Are the cats both males, or females? Are they both neutered or spayed? Sometimes the first cat can get jealous, but try to keep them in separate rooms for a while, and one thing you might try is to give something called "pet sedate" to them both. You can order it online, and it is not expensive. It is made by Francodex, and I have used it for my cats and it does work well. It is homepathic, and herbal, so it will not harm your cats.
Also, it would be best to leave your second cat in your bathroom, or somewhere that the cat can see under a door, or reach under with its paw, and the first cat you have might smell it and play with it. You can usually get them to adjust to each other over a period of several months, but then again, some cats will never get along.
Are they indoor only? That is best if they are, so that they do not have other smells on them from outside.
Make sure you also have separate litter boxes, maybe two for each, and that they have access to the boxes without the one bothering the other. You could also feed them at separate times, and maybe offer treats to them or catnip is always usually good to use.
Let us know if you have further questions or concerns.
God bless and good luck!