Day 1: Delhi to Manali
This probably is the easiest way to do this journey without stressing yourself more than necessary. First day of the journey can easily be about 12-14 hours depending upon your speed so ensure that you take an early start from Delhi; possibly by 3-4 AM. Not much to say about this day except for that this will be a long monotonous day of overtaking truck after truck. Roads are mostly good with a few bad patches in between.
There are several regular buses as well between Delhi and Manali and you will easily be able to find yourself a seat.

Day 2: Manali to Kaza
I am assuming that you would already have arranged for permits; so on this day you can head straight to Kaza. Start early from Manali as well which will be beneficial in two ways. First is that you will be able to avoid a possible traffic jam near Rohtang top. Second is that even though the distance is not that great; the journey from Manali to Kaza can still take a while; thanks to extremely bad roads. From Manali, reach Rohtang top and while descending towards Kokhsar, take the diversion towards Spiti near Gramphoo. It will be a small road on your right but a board marking it as road to Spiti Valley makes it easy to spot.
Also read: Clothes for Spiti Valley
The journey from Gramphoo to Batal is a nightmarish drive. It possibly will be the worst road you have ever been on. After Batal, climb up Kunzum Pass and descend to Losar. From there it is a straight run to Kaza.
There are multiple hotels in Kaza and no matter what your budget is, you will easily be able to find a place to stay. For more information on where you can stay and places you can visit in Kaza, please read Kaza – Sightseeing, Accommodation and Other Facilities
If you were traveling by public transport; then there is an early morning bus from Manali to Kaza which you will have to board. It starts from Manali at around 5 or 6 AM and will see you at Kaza by evening.

Day 3: Kaza to Chandratal
Since we are short on time, the return journey will start the very next day. Destination for today is Chandratal where you will be camping for the night. On the way; you can also pay a visit to Kye Monastery. It is slightly after Kaza towards Losar; right before the bridge where you cross over the river. Reach Kunzum top once again and while descending towards Batal, you will notice a narrow road to your right. This is the 14 kilometer stretch to Chandratal. Road is barely wide enough for one car so drive carefully.
There are several camps at Chandratal of which, Jamaica’s camps is the one that I recommend.
If you reached Kaza by bus than this is where it gets tricky. There is an early morning bus from Kaza to Manali which you will have to catch. By this bus, you will get down at Kunzum pass top near Chandratal diversion from where you have two choices. Either you can start walking for the last 14 kilometers till the lake or stand there and wait for a shared taxi or another tourist to arrive who can give you a ride.
My suggestion will be that you start walking immediately and hitch a ride if someone passed you by. In peak tourist season, lot of people travel this way and you will have no problem hitch hiking.

Day 4: Chandratal to Manali
If you were by your own vehicle than return to Manali via Rohtang pass. If you were however by public transport, than you will have to ensure that you reach back Kunzum Pass slightly before the time you got down here the day before. The Kaza to Manali bus is a daily service and you will have to be in time to catch it once again from Kunzum top.

Day 5: Manali to Delhi
Last day of the trip and nothing but a long journey home.
A lot of people try to squeeze in Chandratal on Day 2. Plan changes to Manali to Chandratal and then Chandratal to Kaza which I would advise highly against. Go straight to Kaza first to avoid any unnecessary hassle due to acute mountain sickness and then return to Chandratal the next day.