Barham joins the Bees on the back of a terrific season where he scored 37 goals alongside ten assists and finished as the top scorer in the Bostik League South.
The 22-year-old has attracted interest from throughout the football Pyramid thanks to his performances in recent seasons.
He started the 2017/18 campaign at Pheonix Sports and was scoring regularly.
He joined league rivals Greenwich Borough in the autumn and he never looked back.
His career began in the youth system at Chelmsford City initially as a right-back but a move to Heybridge Swifts also brought a change of position and he soon found himself up front and in half a season at Heybridge he had already scored over 20 goals.
Barham then joined Tilbury before promotion-chasing Thurrock came in for him and he scored 15 goals as the Fleet went up to the Isthmian Premier.
This will be his first venture into professional football and he's looking forward to making the step up.
He told Barnet`s official website: "It's been a long season, but I've worked hard for this so I can't wait to get started!
"It's nice to get a deal over the line, I can now get my head down into full-time training and work really hard and hopefully do it at this level as well.
"My dream as a little boy was to have my name on the back of a shirt, so to now have that I can kick on and show the Barnet supporters what I can do."
The striker is certainly no stranger to manager John Still, who is pleased to bring him to Barnet.
"I watched quite a bit of Jack last season and I've seen him around the circuit as he's grown up. He's always scored, he's very busy, very enthusiastic and this season he's come on quite a bit.
"Our scouting network picked him up and we really liked him. With me arriving here, it gave me an opportunity to bring him here.
"I think supporters will like him because he has got that non-stop action, he won't stop working and his record of scoring goals is very good. I hope we've found a young player here that will progress, go onto greater things and score lots of goals for us."
Twenty-three-year-old Alexander made 146 appearances for Aldershot Town after initially signing on non-contract terms in February 2015 and joins the Bees for an undisclosed fee.
Alexander started his career at his boyhood club West Ham United, spending ten years in the academy.
After leaving the Hammers, Alexander spent one year at Port Vale, and impressed in a loan spell at Ilkeston Town from December 2013.
Released by the Valiants in the summer of 2014, he joined Concord Rangers in September 2014 and moved onto fellow Conference South side Bishop’s Stortford two months and caught the attention of Aldershot following many good performances and he signed for the Shots in 2015.
Alexander is looking forward to the season ahead: "I'm happy to be here, it's been a long time getting to this point but I'm happy.
"It's a new challenge at a new club and I've got to improve myself and hopefully I can improve the team too."
Still added: "I have known Cheye for a number of years, I've tried to sign him before and it hasn't worked out. He's a right-sided defender and supporters will love him.
"He's good going forward and he's got a great attitude, great work ethic and he's had a couple of good years at Aldershot and has played for the national team as well.
"He's at a great age and he's got a drive in him, he's still young enough to really go on beyond this level and he'll be a terrific addition to this club". |