Aline was the eldest daughter of Hugh Despenser the elder and Isabella Beauchamp. Her actual date of birth is not known, nor is whether she was born before or after Hugh Despenser the younger, the oldest son. She was, however, the first to be married, and while that is not a guarantee of precedence, it does make it more likely that Aline was Hugh and Isabella's first child. Kathryn Warner in her post about Hugh Despenser's siblings here comes to much the same conclusion. Hugh and Isabella probably married in early 1286, which gives a possible year of birth being 1287, with Hugh jnr. being born in 1288/89.
By the time she was about ten, her father had already started to try and obtain a good marriage for her. There is a mention in the Patent Rolls dated January 1st 1296 of the ratification of a transfer of the marriage of the heirs of Philip Burnell to Hugh Despenser senior . Philip Burnell was the nephew and heir to the powerful Robert Burnell, the former Bishop of Bath and Wells and Chancellor of England. When Robert died in 1292, Philip inherited a vast estate, including around 82 manors. Unfortunately Philip didn't live much longer to enjoy them, dying in 1294 in serious debt, aged 29. Edward's mother was Maud Fitzalan, sister to Richard Fitzalan the earl of Arundel. His sister Maud, or Matilda, was named after her.
Philip Burnell's death left Edward Burnell a ward, primarily of the king, as he was far too young to inherit his father's estates. It appears that Edward I then gifted the wardship (and marriage) to the earl of Pembroke, who in turn, as noted above, appeared to have transferred it to Hugh the elder, presumably for a fee. That should have been the last of it but in May 1302 comes another (puzzling) officially recorded grant:
Letters patent of Anthony, bishop of Durham, being a grant by him to Sir Hugh le Despenser, for 1000 marks, of the marriage of Edward, son and heir of Sir Philip Burnel, for the purpose of marrying him to Alina, Sir Hugh's eldest daughter. 
So it must be assumed that something went wrong with the previous grant, that the wardship somehow became the property of Anthony Bek, and Hugh the elder spent a frustrating six years trying to get his hands on the prize he so wanted for his daughter. And seeing as Hugh senior had finally got what he wanted, it must also be assumed that the marriage took place soon after this date.
Edward finally came of age in 1307, doing homage to Edward for his lands on December 6th 1307 . In December 1311 he was summoned to parliament for the first time and became known as Lord Burnell. He seems to have been active in the wars against Scotland and was therefore most likely at Bannockburn. However, the following year tragedy struck the Burnell line again when he died, like his father, aged 29. Even worse, the couple was childless.
Edward Burnell's sister, Matilda, now became his sole heir. She had married John, Lord Lovel of Titchmarsh at some point before 1312, but he was killed at Bannockburn in 1314. This meant that when her brother died, she suddenly became a very attractive marriage partner - and she didn't stay on the shelf for long. She had married again by 4th December 1315 and her new husband was Sir John de Haudlo, a very loyal retainer of, you've guessed it, Hugh Despenser the elder! For more about this 'arrangement', and also the subsequent remarriage of her mother, read Kathryn Warner's article about them here.
But back to Aline. Although most of her late husband's wealth now went to his sister and her husband, Aline still retained her dower manors - those she'd held jointly with Edward. She was still reasonably wealthy and could have made another profitable marriage but it seems that she never married again. The only plausible reason for this is that she did not wish to and so probably took a vow of chastity - the only way an unmarried woman could hold on to her land in those days.
In 1321, her father granted her the manor of Martleye (Martley) in Worcestershire and she took possession without licence from the king. Although she was called to account for it, she was never fined and was actually pardoned (hardly surprising considering it was one of Edward's most trusted friends and counsellors who had given it to her!) . Martleye was one of her manors attacked by Malcolm Musard probably sometime between May 1321 and December 1323. Musard had once been a friend to Despenser senior, and was the previous owner of Martleye. He had wavered in his allegiance during 1318 by siding with the earl of Lancaster, but by 1321 seemed to be firmly back in the royal stable again. He was even appointed keeper of Hugh the younger's castle of Hanley in 1321 and sent to arrest rebels in 1322. However, by 1323, it appears that Edward suspected him of being on the wrong side again and he was arrested in December of that year. Musard was imprisoned in the Tower of London for a couple of years.
By August 1326, when things were starting to go badly for Edward, Musard was pardoned for:
His adherence to the rebels and of the outlawry in the county of Worcester published against him while he was in prison on that account, for non-appearance before the king to answer touching a plea of trespass of Alina Burnel, on condition that he surrender forthwith to gaol, and stand his trial if the said Alina will proceed against him; with restitution to him of his lands taken into the king's hands on account of such adherence. 
The beginning of the year of 1326 saw Aline appointed as the constable of Conway Castle in Wales . This was an incredible honour to be bestowed upon a woman: the only other woman to be so awarded was Isabella de Vesci with the keeping of Bamburgh Castle in 1304. No doubt her brother and father had some influence upon Edward's decision, but why it was thought of as a good idea in the first place is a mystery. Sadly for Aline, it did not last long. With the invasion of Isabella and Mortimer and the need for Edward and Hugh to flee London, she was replaced as constable by her steward, William de Ercalewe (another man very loyal to the Despensers and to Edward II).
The death of her brother and father and the deposition of Edward must have made Aline fearful for her own future. But remarkably she was left untouched by the new Isabella/Mortimer regime, as was Ercalewe and de Haudlo. Maybe as a sign of her gratefulness, or maybe as a way of escaping the pressure cooker that the country was becoming under the new rule, she decided to go on pilgrimage to Santiago - the place her father had gone in 1319. Three grants of protection were issued in November 1329, April 1330 and finally in February 1331. Hopefully, after the false starts, she managed to make it to Spain and pay her respects.
She seems to have had a peaceful life during the reign of Edward III. In 1338 she was granted alienation in mortmain for some pieces of land to give to two chaplains who were to celebrate a daily service in the chapel of St Giles at Lolleseye (Lulsey) in Worcestershire. Prayers were to be said for the souls of her late husband, her brother, as well as her nephew (Hugh the even younger), the faithful William de Ercalewe and Walter de Lench, a neighbouring landowner . Interestingly her father was not included amongst this list of men who she had great fondness and love for, nor were any other members of her family.
According to the Fine Rolls and an IPM, Aline died on 16th May 1363, well into her seventies . With no heirs, her dower manors reverted back to the heir of her sister-in-law: Nicholas Burnell.
 Calendar of the Patent Rolls 1292-1301, p.179
 A Descriptive Catalogue of Ancient Deeds, Volume 4, No. A.6278
 Calendar of the Close Rolls 1307-1313, p.11
 Calendar of the Patent Rolls 1321-1324, p.30
 Calendar of the Patent Rolls 1324-1327, p.304
 Calendar of the Patent Rolls 1324-27, p.215
 Calendar of the Patent Rolls 1334-1338, p.50 Calendar of the Fine Rolls 1356-1368, p.277, and Chan. Inq. p. m. 37 Edw. III (1st nos), 14 (C 135/177/12)