Eucalyptus and other aromatic oils can bring great relief to difficult breathing or a congested chest, whether it's due to a cold, flu, bronchitis or asthma.
To get the maximum effect, place a few drops of the oil in a basin of hot water. Put a towel over your head and lean forward over the bowl. The more of a "tent" you can make with your towel to enclose the steam, the more concentrated the effect. This gives you a double benefit - the clearing effect of the oil plus the soothing effect of the moist steam.
Take care, though, especially with young children or the elderly, who could scald themselves. Use a very stable basin and don't leave them alone. Some people may find this method uncomfortable, because the aroma can be too strong, or the perfumed steam stings their eyes. If this applies to you, there are other ways to get a similar benefit.
One option is to take a bath, and add a few drops of the oil while the water's running. The whole bathroom will fill with aromatic, soothing steam, and the bath itself will be relaxing.
Other alternatives are to put the oil in a vaporizer (available from your pharmacy), or an oil burner (just the usual kind you would use for essential oils). In each case, follow the instructions - in some oil burners you use the oil neat, whereas in others you have to add some water.
For night time, if you don't like leaving a candle burning, you could try putting 4 or 5 drops on the pillow. I don't find that particularly effective - unless you sleep in one position all night long! You can also put drops on a handkerchief, but be careful - if you let it touch a sensitive nose, it will sting! Similarly, you can rub the drops directly on to the chest, but do try a small area first, because many people find it irritates their skin.
If you can't get eucalyptus oil, there are several alternatives that you can use, including some combination oils. Depending on where you live, you might be able to find Friar's Balsam, Olbas Oil, cedarwood, clove, pine, or tea tree oil. You'll also find some chest rubs which contain these oils.
Bear in mind that eucalyptus oil is poisonous if swallowed, and therefore inhaling it over a prolonged period can also be unhealthy, so use it wisely. In any case, if you find you need to clear your chest frequently, you should get yourself checked out - you may have asthma or bronchitis, which need proper medical treatment. If you have a heart condition of any kind, it's best not to do steam inhalations without checking with your doctor first.