First of all; this is a high altitude climb and the approach has to be done in steps, if not acclimatized.
Copiapó, a quite large city, is the starting point from the Chilean side and the only feasible way to travel towards the highlands is by car. There is no public transport linking Copiapó with the Ojos del Salado area.
There are some options how to get there.
1). Join a commercial expedition/package tour.
2). Rent or charter a vehicle.
If you opt for alternative 3, be sure you bring lots of water and you have to be prepared for long waits as there are very little traffic on this remote road.
Commercial expeditions to Ojos del Salado usually travels from Copiapó to Laguna Verde, where they stop for a couple of days of acclimatization.
If driving your own car, head east on Chilean highway # 31 towards Paipote and Paso de San Francisco. After 10 km you take a left for Paipote and another 15 km take a right and follow the 31.
There's a turn-off after about 80 km and you'll have a choice of two roads leading to the border post. These two roads are quite similar and it doesn't really matter which of them you go for. About 4 hours out of Copiapó you arrive at the border post, where you officially leave Chile. This post is located about 180 km from Copiapó and the officials who stay there can usually update you about the weather and snow conditions on the peak.
The distance Copiapó to Laguna Verde is roughly 270km and that's about 6 hours in a car.
Laguna Verde is located at 4300 meters and it's a very good idea not to go all the way there in one go. You will feel the altitude and if you're unlucky, you're hit by altitude sickness and have to go down the way you arrived.
There's a refugio at Laguna Verde. You can store gear here and possibly also use the kitchen facilities. Fresh water is sometimes readily available here, but you may have to go and get it yourself. It's to be found past the police post. Sometimes the climbing permits are checked here. If in doubt where to find the water, ask the police.
A rough road takes you to Refugio Atacama at 5200m. The ride takes about 1 hour and a quarter and has to be done by a jeep or a 4 wheel drive. Hitching this stretch is difficult as the very few jeeps often are full of climbers.
The climb, if you can call the initial low grade slopes as such, starts from Refugio Atacama. A well defined walking path leads upward and it'll take you to the next refugio - Refugio Tejos at 5900 meters. This walk takes anything from 1½ to 4 hours, much depending on how acclimatized you're.
Refugio Tejos is the last hut along the route and from here it's all camping if you need more stops before the summit.
The trail is usually very easy to find, but if there has been snow or you for any other reason have difficulties, head right diagonally upwards, underneath the foot of the hulking shape of the mountain. When you have reached a point where you have the summit block right over you, look for paths leading more directly up. There may be more than one choice here, but they all later join into one trail further up.
The climb flattens out a bit when reaching a sloping plateau underneath the summit block. It's very clear where the highest point of the ridge in front of you is, but don't head straight for it. The scree, rocks and sand are much worse to approach head on. You better head towards the left high point on the ridge and when reaching the scree, start an upwards traverse in direction of the summit block.
You are now in a small rock coloir and just follow it up to the gap in between the double summit. When at the gap, you either shoot for the "Chilean Summit" on your right or the "Argentinean" on the left. There has been some debate about which is the highest, but most consider the Chilean summit slightly higher. A Brazilian expedition brought two black, metal pyramids to place on the both of the summits, just to make sure one of these ended up on the highest spot.
Anyway, there are ropes hanging from both summits and I would be very careful about using them. Some of them seem to have been hanging there for years and years.
The last meters to the summit (whichever of them you choose) are on semi-technical rock. On sea level it would've been a breeze, but up here on almost 7000 meters, usually pounded by gale-force winds, it can be hard. Be very careful as a fall from any given spot up here will cause a lot of damage, possibly death. The rock is rotten and not to be trusted.
Many commercial organizers may tell you it's a 100% straight forward climb. It is, to the summit block, but from there on you have to be prepared for some tougher scrambling.
When you have reached the summit, the reward is great!
The fantastic scenery of the Puna is all around you, with its crystal clear blue lakes, the many snow capped volcanoes and the desolate plains all around you, far below.
Under normal conditions, neither ice ax, nor crampons are necessary, but going in the beginning or the end of the season, they may both by necessary.
I would bring both as blizzards can be fast and hard here.
A short length of rope may come in handy, unless you really trust your scrambling/climbing skills up high.
Be prepared for the worst at all times in this area, even if going in the summer season (late November to late March). The weather can grim.
Bring lots of water as there are none higher up on the peak and even ice and snow can be difficult to find sometimes. Have a good look at the peak to find out how much there are on the peak before setting off.