The distance is 4km (less than 3 miles). At that distance, you don't need any special clothing and you don't need to spend lots of money to get started. Keep your speed down so you dont sweat. Use your gears, esp the low ones, so you are pedalling at about 80rpm but with low pedalling force. You just wind yourself up a hill.
Fit some full-length fenders to your bike to keep the dirty roadsplash off your clothes. A 5C, you can wear breathable waterproof overpants in heavy rain.
Get some lights, some bright jackets, a windproof one and a waterproof one.

For my short rides around town in similar conditions, I usually wear casual hiking style pants in polyester cotton. I have worn cotton T shirts but synthetic wicking ones are more comfortable and expensive Merino ones are the very best. You can pick up polyester T shirts for a couple of dollars.
Between the base layer and the waterproof or windproof shell, it doesn't matter too much what you wear. I like polycotton polonecks and wool sweaters.
Use a waterproof light hiking boot or show with wool socks.

If you are relying on cycling to get you to work, invest in some quality tyres with good puncture protection (eg Schwalbe Marathon). You have get super protection tyres like Marathon Plus, they are more expensive but you can ditch your puncture repair kit.
The repair kit should be a spare innertube, tyre levers and any wrench you need. I rarely take a repair kit for journeys under 3 miles.