Plumbing – this bathroom basin tap connection and can I replace it theself

Tighten the joint slightly.

Pipe threads are tapered - the PTFE thread tape (or pipe dope - I'm currently more of a fan of PTFE pipe dope than of PTFE thread tape, though I used to be a big fan of the tape as opposed to old-style dope) provides a seal for the "helical leak path" (the fact that the female threads are a bit sharper than the top of the male threads) and some lubricity for easier tightening.

A slow drip type leak is often just a 1/4 turn from not leaking.

If you take it apart, you need to clean off the remains of the old tape and apply new tape (one reason I now prefer the modern dope - you can just add a bit more, with no need to clean off the old, and you don't have all the tape shreds to deal with.) Excessive thickness of tape won't help the seal - more is not better.

The original leak is most likely either caused by a failure in the rubber washer that connect your supply line to your fill valve, or a leak in the seal that connects your fill valve to the tank.

This is a common problem, especially with older toilets that use natural rubber washers. The rubber breaks down over time and will eventually begin to leak. The newer synthetic washers will last much longer.

It is unfortunate that you already have taken it apart because you will have a harder time determining the source of the leak. When you run into a leak of this kind, it is best to first check the supply line itself, as it is the easiest to access. Shut off the water supply, and drain the tank of water. Then unscrew the supply line and inspect the washer. It should be firm and flexible, yet hold its shape and be in one piece. If it is not, then it needs to be be replaced and you can reinstall the supply line. After a quick test, you should be able to see if this fixed the leak. If it leaks from the supply, make sure that the washer is in place. On Occasion, the washer is missing right from the beginning. They either fall off, or people take them off.

The other place it could be leaking is from inside of the tank. You will need to disassemble the toilet as above. You will then needs to loosen the retaining nut on the outside bottom of the tank where the supply line goes. Loosening this nut should allow you to lift out the fill valve. Inspect the washer on the valve, and check if it is good. Also check the threaded part to make sure there aren't any cracks in the fill valve body as well. Replace washer, and reinstall.

By replacing both of these washers, it should solve the leak. If it doesn't then there may be a fault in the parts themselves - either the supply kine is bad, or the fill valve, or possibly a combination of both. Also be sure that it is leaking from the supply, and not from anywhere else. There could either be a crack in the porcelain, or there could be a bad tank to bowl seal. There are "universal" rebuild kits available out there that include everything that you need to rebuild a toilet. They include all washers, fill valve, and a flapper. The only part that is not included is the supply line. If the internals of the toilet are old, or not functioning properly, then it would be a good idea to do an entire rebuild.