Here is a pretty quick explanation.
Starting from where the water will enter the drain and ending where the water will flow out, dig a ditch that's about 10 inches deep and 6 inches wide. Keep in mind that you will need to dig deeper if your starting point slopes upward.
Lay about 2 inches of rock into the trench before placing the pipe lengthwise into the ditch.
Cover the pipe with another layer of rocks, again about 2 inches.
Fill in the remaining space with soil and then plant some grass.
Here is a pretty good video, which also shows it's a good idea to add a clean-out.
They also say you should not use flexible corrugated perforated plastic drain pipe, because it cannot be cleaned like rigid pvc can be.
Your catchment and cistern (the part that's open to air and thus to contamination by insects/bacteria) should be as small as it can be while still providing enough capacity to take in the rain as it falls. Once rain enters the cistern from the catchment, it should then be pumped into a holding tank which is air- and water-tight, and from which you draw your water for general use. That way, water doesn't stagnate in the cistern. To make this work properly, you'll want a drain in the cistern. Normally the drain would be kept open to prevent standing water in the cistern between rains. When rains start, the first few minutes' worth should "wash" the catchment and cistern relatively clean of contaminants like dust, pollen, bird doo, etc; then the drain can be plugged and water diverted into the holding tank. You can do this manually, or set up some sort of rain-sensing system that will control the drain for you.
Once water is in the holding tank, a small amount of antibacterial and algaecide agents will keep it clean. Iodine kills both bacteria and algae, and you need to take in a little of it anyway (good for the thyroid), but the water will smell a little funny, will not be suitable for laundry (iodine permanently stains cotton and most fabrics so over time your whites will become yellow) and too much iodine can be poisonous. Chlorine is a no-brainer antibacterial but is less effective against algae except in large (harmful) doses, and will also make for funny-smelling water.
You could consider a saltwater chlorination system. These systems use a small amount of sodium chloride (3.5g/L is the threshold for human perception of salt by taste; saltwater systems generally work with salt concentrations between 2-3g/L) along with electrically-charged plates to create free chlorine ions that disinfect the water. Saltwater chlorination is used for pools to avoid chloramines ("combined" chloride compounds, which give the water that distinctive smell) and similar systems are available for producing potable water.
Understand that any chlorine-based disinfecting system can produce "disinfection by-products" that can be harmful; however these are mostly a concern in large pools where lots of people are shedding organic material (bacteria, dead skin, etc) that forms the raw materials for these by-products. Rainwater that you're not swimming in, and that you're actively keeping new contaminants out of, shouldn't have this problem. Despite that, you can also consider a simple carbon filter coupled with a UV sanitizer; these systems are also commercially available for water sanitization in systems like yours.
Lastly, you'll need to keep the bottom of the tank relatively clear of sediment. The easiest method in an above-ground tank is to have a drain at the bottom which you can open for a short time to flush sediment. The next-easiest way is to vacuum it, just like you would a pool. Every few years you may want to drain the thing completely at the end of the dry season and give it a scrub, to clean off scaling and other build-up.
Best Answer
If the existing surface preparation is sloped the wrong direction and water runs to an undesirable area then you would normally work to re-install the surface preparation to get a different slope.
Since you have a desire to not replace your tarmac due to cost considerations then your only real choice is to do one of the following:
Install an additional curbing at the lower edge of the tarmac area to divert the runoff water to another area. The height of curbing required would be fully dependent on conditions at your site.
Install a French style drain at the area adjacent to the tarmac. This drain would be a pit dug into the ground that can accept the run off water and allow it to disperse into the soil. Generally such pit is filled with rocks or course material that can let water enter easily into the pit. It is often good to cover the pit with a screen and lid. The screen can be used to filter organic material from entering the drain and hopefully preventing it from plugging up.
You could install a sump pit with a water pump unit. The pit would have an opening at the low spot where water collects so that the water can enter the sump pit. A water level switch on the pump system turns on the pump when the a certain level of water has accumulated. The pump moves the water out of the sump pit through a drain pipe that is plumbed to an area where the water can be safely discharged. This could be to a lower area of your property, out to the street curb or into a storm sewer system. This solution also requires the availability of electrical service to the pump area.