I've written four posts about Henry of Lancaster's brother Thomas, three about his son Henry of Grosmont, and one about his daughters, but never about Henry himself, so it's high time I rectified that (thanks to reader Kate S. for the the suggestion!).
Henry of Lancaster, Edward II's first cousin, was born in or about 1281 as the second son of Edward I's brother Edmund, earl of Lancaster, Leicester and Derby (16 January 1245 - 5 June 1296), and Blanche of Artois (c. 1245/48 - 2 May 1302). Blanche was the niece of Louis IX of France, queen of Navarre by her first marriage, mother-in-law of Philip IV of France and grandmother of Louis X, Philip V, Charles IV and Isabella of France. (Yes, Isabella's grandmother was also the aunt by marriage of Isabella's husband Edward II; oh, these complicated royal inter-relations.) Henry of Lancaster was thus grandson and nephew of kings of England, brother-in-law of a king of France, half-brother of a queen of Navarre, and uncle of three kings of France and a queen of England. So was pretty well-connected, then.
Sometime before 2 March 1297 , when he was about fifteen, Henry married the heiress Maud Chaworth (elder half-sister of Hugh Despenser the Younger), who was born on 2 February 1282 and inherited the lands of her father Patrick and uncle Payn in Carmarthenshire, Glamorgan, Hampshire and Wiltshire. Henry inherited a part of his father's vast lands, though of course the bulk of it went to his elder brother Thomas, and was lord of Kidwelly and owned the Three Castles in Monmouthshire (Grosmont, Skenfrith and the White Castle). Henry and Maud had six daughters - the countesses of Arundel and Ulster, the prioress of Amesbury, Lady Percy, Lady Wake and Lady Mowbray - and one son, the great Henry of Grosmont, first duke of Lancaster. Henry was the grandfather of: Elizabeth, duchess of Clarence; Blanche, duchess of Lancaster (both of the latter daughters-in-law of Edward III); Richard, earl of Arundel, executed by Richard II in 1397; Thomas, archbishop of Canterbury; Maud, countess of Oxford; Alice, countess of Kent; Joan, countess of Hereford; Henry, earl of Northumberland and Thomas, earl of Worcester; and was also the great-grandfather of King Henry IV and of Philippa, queen of Portugal.
As well as his elder brother Thomas, Henry of Lancaster had a younger brother, John, who died childless in 1317; Henry was heir to his lands in France, including Beaufort, after which the illegitimate children of his grandson by marriage and heir John of Gaunt were named. Henry was widowed sometime before 3 December 1322 , and never remarried (the Alix de Joinville often named on genealogical sites as his second wife was in fact the wife of his brother John). How Henry got on, or not, with his brother-in-law Hugh Despenser the Younger is a matter for speculation, though he seems to have been a loving and caring father to his children and formed close relationships with them: his daughters Blanche Wake, Joan Mowbray and Maud de Burgh (and perhaps the others) lived with him most of the time even after they married, and Henry paid all or most of the expenses of his son Henry long after he reached adulthood. There is also evidence that the siblings were close and that they and their spouses often travelled together.  On the other hand, Henry's relationship with his elder brother Thomas was, according to Thomas's biographer John Maddicott, a distant one  - for which Thomas's prickly personality was perhaps largely to blame.
Given the eminence of the Lancaster brothers, it's surprising how little is known of their early lives; even their dates of birth are not known. Henry accompanied his elder brother and their cousin by marriage Jan of Brabant - husband of Edward II's sister Margaret - on several visits to their cousin the future king in 1293, and were with him for his ninth birthday on 25 April. The boys stayed with Edward again in June, Henry and Thomas accompanied by thirty horses and twenty-one servants and Jan by thirty horses and twenty-four servants, much to the exasperation of Edward's clerk, as Edward's household had to bear all the costs ("They are still here" and "Here they are still, and this day is burdensome," the clerk wrote). In September 1293 Henry stayed with Edward again, for longer than expected, as he fell ill.  Henry accompanied his uncle Edward I on the Flanders campaign of 1297, when he was perhaps only sixteen - he wasn't paid for it - and took part in the king's siege of Caerlaverock three years later; the Roll of Arms of Caerlaverock says of him:
"I may go on to speak of Henry,
Whose whole daily study
Was to resemble his good father..." 
In the Barons' Letter to the pope of February 1301, Henry - then about twenty - is named as 'Henry of Lancaster, lord of Monmouth' (Henricus de Lancastre Dominus de Munemue). He appears tenth on the list, after the eight earls and William, Lord Leyburn. According to the later and frequently unreliable chronicler Jean Froissart, Henry was known as Wryneck (Tort-Col in French). To quote Scott L. Waugh in the ODNB, this name "could refer either to a physical deformity, or to the stiff-necked pride that he displayed in his family's political and territorial heritage."
In February 1308, Henry was ordered to meet his cousin Edward II and Isabella of France on their arrival in England after marrying in Boulogne.  Whether Henry had ever met his niece before, I don't know. At their coronation several weeks later, Henry carried "the royal rod [virga], at the top of which was a dove" in the procession.  He crops up occasionally throughout Edward's reign, though as he was a younger brother, he wielded little political influence and didn't play a huge role in events of Edward's turbulent reign. Edward was probably annoyed with Henry in 1316, when Henry's eldest daughter Blanche (then aged about twelve or fourteen) married Sir Thomas Wake without the king's permission; Edward had been planning to marry Wake, his ward, to his great-niece Joan Gaveston. Still, it doesn't appear to have had a deleterious effect on the two men's relationship, and Henry took part in the campaign against Llywelyn Bren in early 1316, with Sir William Montacute. 
Henry's younger brother John died childless in 1317, and in May 1318 Edward II granted him permission to travel to France to "obtain the inheritance in that land which by the death of John de Lancastre, his brother, descended to him." In June 1319, Edward excused Henry from attending the siege of Berwick as he was "beyond the seas on important business." Henry appears to have spent much if not all of the next few years in France, to judge from the number of times Edward granted him permission and protection to remain overseas (he was still out of England in January 1322 and perhaps even later).  The pope wrote to Henry several times in 1318 as a close kinsman of both the king and the earl of Lancaster, and "bound to pay them reverence and affection," asking him to promote accord between them "so that the realm may be freed from disturbance" (Edward and Thomas of Lancaster were, to cut a very long story short, feuding endlessly).  I wonder how effective Henry's intervention was, assuming he even attempted one, given that he was not close to his brother and had little if any influence over his cousin Edward II.
Henry's life was to change dramatically in March 1322, when his brother Earl Thomas was beheaded for treason. As Thomas had no legitimate children by his wife Alice de Lacy (though had at least two illegitimate sons), Henry was heir to his vast inheritance - but Edward II was unwilling to give all or even most of it up to him, and Henry was to play a vital role in the revolution of 1326/27 and the regime of his niece Isabella.
1) Calendar of Patent Rolls 1292-1301, p. 239 (a grant to "Henry de Lancastre and Matilda his wife").
2) Calendar of Close Rolls 1318-1323, p. 687 (when the "executors of the will of Matilda, late the wife of the aforesaid Henry" are mentioned).
3) Kenneth Fowler, The King's Lieutenant: Henry of Grosmont, First Duke of Lancaster 1310-1361, pp. 26-27.
4) J.R. Maddicott, Thomas of Lancaster 1307-1322: A Study in the Reign of Edward II, p. 319.
5) Seymour Phillips, Edward II, pp. 51-53.
6) Maddicott, Thomas of Lancaster, p. 68 (for 1297); Thomas Wright, ed., The roll of arms, of the princes, barons, and knights who attended King Edward I to the siege of Caerlaverock, in 1300, p. 20.
7) Close Rolls 1307-1313, p. 51.
8) Ibid., p. 53.
9) Calendar of Chancery Warrants 1308-1348, p. 439.
10) Patent Rolls 1317-1321, pp. 145, 146, 153, 217, 329, 343, 503; Ibid. 1321-1324, p. 69.
11) Calendar of Papal Letters 1305-1341, p. 439.