Prayer in Difficulty
Seated, lying, travelling, no water, or in a non-Muslim environment. Islam has a ruling for every situation.
صَلِّ قَائِمًا، فَإِنْ لَمْ تَسْتَطِعْ فَقَاعِدًا، فَإِنْ لَمْ تَسْتَطِعْ فَعَلَى جَنْبٍ
"Pray standing; if you cannot, then sitting; if you cannot, then on your side."
Sahih Bukhari 1117
Praying on a chair is fully valid for anyone who cannot stand or sit on the floor due to illness, old age, or disability. The chair should ideally be placed slightly behind the row to avoid blocking others during your Sujood inclination.
Make intention (niyyah) in the heart for the prayer you are performing. Raise both hands to ear-level and say "Allahu Akbar", you may hold the armrests.
Sit upright in the chair. Recite Thana, Ta'awwudh, Bismillah, then Surah Al-Fateha and another surah, exactly as you would in standing.
Lean your upper body forward about 45° from the waist. Place hands on knees if you can. Recite "Subhana Rabbiyal Adheem" 3 times.
Return to sitting upright. Recite "Sami Allahu liman hamidah. Rabbana lakal hamd."
Lean forward further than Ruku, as far forward as comfortable. If a table or stool is in front, you may rest your forehead on it. Recite "Subhana Rabbiyal A'la" 3 times.
Return to sitting upright briefly. Recite "Rabbighfirli" once or three times.
Lean forward again as in the first Sujood. Recite "Subhana Rabbiyal A'la" 3 times.
Remain seated. Recite At-Tashahhud, then Salawat Ibrahim, then any dua. Complete with Salaam to right and left.
Based on mainstream Sunni scholarly positions. Rulings cited from Hanafi fiqh unless noted otherwise. Consult a qualified scholar for personal situations.