Neem trees are heat loving plants and grow best in hot climates. They can tolerate cooler weather, down to about 35 F or 5°C.

However, if your climate has long periods of such cool weather, or if it gets even colder than that, then you won't have much luck growing a neem tree plant in your garden.
How To Grow A Neem Tree As A House Plant
Neem grows easily from seed. If you can get hold of seeds, that is.

Just put your neem seeds in a pot with a very good quality potting mix, cover them with an inch of the mix, and keep that pot moist and warm. Neem seeds should germinate within one to three weeks.The cooler the temperature, the longer the seed will take to germinate.

Why is a good quality mix so important when neem is such a hardy tree that will grow well even in the poorest soil?

Because if you grow a neem tree outside it can develop a huge root system. Its tap root will reach far, far down to mine the deep layers of soil for nutrients, and lateral roots will spread wide and far to go foraging for more.

A pot is not a natural environment for such a vigorous tree. It is very restricting, so you have to look after your neem house plant a lot better and give it more care. Make sure you use a good quality, free draining potting mix.

Once the seed has germinated and the tree is growing fertilize it regularly with a complete and balanced fertilizer. Adjust the amount to the size of the tree and the time of the year. Use more when it is actively growing, cut back or stop fertilizing in winter.

As a guideline, while the tree is actively growing, I would suggest weekly watering or spraying with dilute fish emulsion (use about half of what the package says), and a monthly application of a balanced solid organic fertilizer.

Your neem house plant will initially grow only slowly, but once it is a year old it should take off, provided you look after it well enough.