Soorya Vansh - Lord Rama family tree - from Brahma onwards

Rama's family history as narrated by Guru Shri Vashishtha at the time of the wedding with Seetadevi

1. Brahma,
2. Mariichi
3. Kaashyapa is the son of Mariichi,
4. The Sun
5. Manu
6. Ikshvaaku is the son of Manu,
7. Kukshi,
8. Vikukshi is the son of Kukshi...
9. Baana
10. Anaranya is the son of Baana.
11. Pruthu is the son of Anaranya
12. Trishanku is Pruthu's son
13. Dhundumaara happened to be the son of Trishanku
14. Dhundumaara begot Yuvanaashva as son
15. Mandhaata emerged as the son of Yuvanaashvas
16. Susandhi
17. Susandhi engendered two sons, namely Dhruvasandhi and Prasenajit
18. From Dhruvasandhi Bharata is begotten.
19. Bharata begot a highly effulgent son named as Asita
20. Asita, waged war with kings like Haihaya-s, Taalajanghaa-s, and the valiant Shashabindu-s. While counterattacking those kings, Asita is dethroned in war and then he reached Himalayas along with his two wives. At the time of his demise two of his wives were pregnant. Kalindi gave birth to a son. He took birth along with the poison administered to his mother, he became Sagara, the emperor

21. From Sagara it is Asamanja
22. From Asamanja it is Amshuman
23. Amshuman's son is Diliipa
24. The son of Diliipa is Bhageeratha
25. Bhageeratha son is Kakutstha
26. Kakutstha's son is Raghu.
27. Raghu's son is the great resplendent Pravriddha also known as Kalmashapaada
28. Pravriddha, Shankana
29. Shankana's son is Sudarshana
30. Sudarshana it is Agnivarsna
31. Shiigraga is the son of Agnivarsna
32. Shiighraga's son is Maru
33. Prashushruka
34. Ambariisha is the son of Prashushruka
35. Ambariisha's son was Nahusha
36. Yayaati is the son of Nahusha
37. Naabhaaga is born to Yayaati
38. Aja was Naabhaaga's son
39. From Aja,  Dasharatha is manifest
40. From this Dasharatha, these brothers, Rama and Lakshmana
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Another source is here - http://en.wikipedia.org/wiki/Genealogy_of_Rama

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Another description (tabular)is in the Bhagwatam.

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Another description is here (Slightly more elaborate and detailed) :

1. Brahma created 10 Prajapatis , one of whom was Marichi.
2. Kashyapa is the son of Marichi and Kala. Kashyapa is regarded as the father of humanity. His sons from Aditi, the sky goddess, and the daughter of Daksha Prajapati are called Adityas - 
4. one of whom is 
 Vivasvat or Vivasvan .
5. Vivasvan or Vaivasvata - the Sun God
6. Manu or Vaivasvata Manu – the King of all mankind and the first human being. (According to Hindu belief there are 14 Manvantaras; in each, Manu rules. Vaivasvata Manu was the seventh Manu . Manu had nine sons, Vena, Dhrishnu, Narishyan, Nabhaga, Ikshvaku, Karusha, Saryati, Prishadhru, Nabhagarishta and one daughter, Ila. He left the kingdom to the eldest male of the next generation, Ikshvaku, who was actually the son of Manu’s brother Shraaddev.
7. Ikshvaku – the first prominent monarch of this dynasty, giving the dynasty its another name the Ikshvaku dynasty.
8. Vikukshi – He is said to have eaten the meat of a rabbit at the time ofShraddha and was known as Shasad. (Some records claim him to be grandson of Ikshvaku.) His son was Kakuthsa or Puranjay.
9. Kakutstha or Puranjaya – He was a brave king and fought in the Devasur Sangram. His original name was Puranjaya. He was also known as Kakuthstha, which means seated on the hump. His dynasty was also known as Kakuthstha after him.
10. Anena or Anaranya
11. Prithu
12. Vishvagashva
13. Ardra or Chandra
14. Yuvanashva I
15. Shravast – He founded the town of Shravasti near Kosala.
16. Vrihadashva
17. Kuvalashva – He killed a Rakshasa named Dhundh. It is said that Dhundhar region and the Dhund river are named after Dhund and  was called “Dhundhumara”.
18. Dridhashva
19. Pramod
20. Haryashva I
21. Nikumbh
22. Santashva
23. Krishasva
24. Prasenjit I – His daughter Renuka was married to sage Jamdgni. She was mother of Parashurama.
25. Yuvanashva II – He was married to Gori, daughter of the Chandravanshi king Matinaar.
26. Mandhata – He became a famous and Chakravarti (ideal universal ruler) king. He defeated most of the other kings of his time. He married Bindumati, a daughter of the Chandravanshi king.
27. Ambarisha – Great devotee of Vishnu.
28. Purukutsa & Harita – Purukutsa performed the Ashwamedha Yajna (horse sacrifice). He married Nagkanya “Narmada”. He helped Nagas in their war against the Gandharvas. Harithasa gotra linage starts from here.
29. Traddasyu
30. Sambhoot
31. Anaranya II
32. Trashdashva
33. Haryashva II
34. Vasuman
35. Tridhanva
36. Tryyaruna
37. Satyavrata or Trishanku – His original name was Satyavrata, but he committed three (tri) sins, and hence got the name Trishanku. Trishanku also had a desire to ascend to heaven in his mortal body. Vashistha refused him this boon, since it is against nature to ascend into heaven as a mortal, the sage Vishwamitra, Vashistha’s rival, created another heaven for him, called “Trishanku’s Heaven”, and located in mid-air. 
38. Harishchandra – He is known for his honesty, truth and devotion to duty or Dharma.
39. Rohitashva – He was the son of Harishchandra. He founded town of Rohtas Garh in Rohtas district, Bihar and Rohtak, originally Rohitakaul, meaning from the Kul (family) of Rohit
40. Harit
41. Chanchu
42. Vijay
43. Ruruk
44. Vrika
45. Bahu or Asit – He was attacked and defeated by another clan of Kshatriyas. After this, he left Ayodhya and went to the Himalaya mountains to live as an ascetic with his queens. At that time Yadavi queen was pregnant with Sagara.
46. Sagara – He recaptured Ayodhya from the “Haihaya” and “Taljanghi” Kshtriyas. He then attempted to perform the horse sacrifice, Ashwamedha Yajna, but the sacrificial horse was stolen by the god Indra on the south eastern shores of the ocean, which was at that time an empty bed with no water in it. At least sixty of Sagara’s sons died attempting to recover the horse, also causing great destruction by their reckless search. Puranic legends say the number of his sons was 60 thousand.
47. Asmanja – Sagara’s surviving son was not made king due to his bad conduct.
48. Anshuman – He was the grandson of Sagara, and his successor as king. He did penance in an attempt to bring the holy river Ganges to earth, that she might wash away the sins of his ancestors.
49. Dileepa I – He also tried to bring Ganges to earth, but also failed.
50. Bhagiratha – Sagara’s great-grandson, after strenuous penances, at last succeeded in bringing Ganga down from heaven. When she flowed over the remains of his ancestors, their souls were redeemed, and the ocean was refilled. Ganga also bears the name “Bhagirathi”, in honour of his deed.
51. Shrut
52. Nabhag
53. Ambarish – According to Buddhist legends, he went to Tapovana to be a renunciant but after a public outcry returned and ruled for some time.
54. Sindhu Dweep
55. Pratayu
56. Shrutuparna
57. Sarvakama
58. Sudaas
59. Saudas or Mitrasah – He performed the Ashwamedha Yajna, but as the rituals were concluding a Rakshasa tricked him into serving human meat to Brahmin,s including Rishi Vashishta. He was then cursed by the Brahmins. He wanted to curse them back, but his wife prevented him. He spent twelve years in exile in the forest.
60. Sarvakama II
61. Ananaranya III
62. Nighna
63. Raghu I
64. Duliduh
65. Khatwang Dileepa
66. Raghu II or Dirghbahu – He was a famous king, who conquered most of India. The great epic Raghuvamsa describes his victories. After him the Sun dynasty was also known as the dynasty of Raghu as Raghav (Raghuvanshi).
67. Aja
68. Dasaratha
69. Rama – He is considered the seventh Avatar of the god Vishnu. He is worshiped by every Hindu. Many Hindus include his name in either their first or last name. Rama’s story before he became king of Ayodhya is recounted in the Ramayana. After he ascended the throne, he performed the Ashwamedha Yajna. Bharata, his younger brother, won the country of Gandhara and settled there.
70. Lava and Kusha – They were the twin sons of Rama and his wife Sita. Lava ruled south Kosala while Kusha ruled north Kosala, including Ayodhya. 
71. Kusha married “Nagkanya” “Kumuddhati”, sister of Kumuda. After Kusha the following kings of the solar dynasty ruled Ayodhya:
72. Atithi
73. Nishadh
74. Nal
75. Nabha
76. Pundarika
77. Kshemandhava
78. Dewaneek
79. Ahinagu, Roop and Rooru
80. Paripatra
81. –(unknown name)
82. Bala
83. Ukta
84. Vajranabh
85. Shankh
86. Vishvashaha
87. Hiranyanabha
88. Pusya
89. Dhruvsandhi
90. Sudarshan
91. Agnivarna
92. Shighraga
93. Maru
94. Prasut
95. Susandhi
96. Amarsha
97. Vishrutwan
98. Vishravbahu
99. Prasenjit I
100. Takshaka – Laid the foundation of Nagavansh
101. Brihadbal – He fought in Battle of Kurukshetra on the Kaurava side and was killed in battle.
102. Brahatkshtra
103. Arukshay
104. Vatsavyuha
105. Prativyom
106. Diwakar
107. Sahdeva
108. Vrihadashwa
109. Bhanuratha
110. Pratitashwa
111. Supratika
112. Marudeva
113. Sunakshtra
114. Antariksha
115. Sushena
116. Anibhajit
117. Vrihadbhanu
118. Rawats
119. Dharmi
120. Kritanjaya
121. Rananjaya
122. Sanjay
123. Prasenjit II – He was a contemporary of Gautama Buddha and King Bimbisara of Magadha. His sister, Koushala Devi, was married to Bimbisara. The city of Kashi (Varanasi) was given as a dowry to her. After Bimbisara was murdered by his own son Ajatshatru, Prasenjit undertook a long series of wars with Ajatshatru. He also respected Buddha, who was also a Kshatriya from solar dynasty. In Buddhist literature he is addressed as “Pasenadi”.
124. Kshudrak
125. Kulak
126. Surath
127. Sumitra – He was the last king of Ayodhya from solar dynasty. In the fourth century BC, emperor Mahapadma Nanda of the Nanda Dynasty forced Sumitra to leave Ayodhya. He went to Rohtas with his sons. His son Kurma established his rule over Rohtas.